The electrical conductivity of the water depends on the water temperature : the higher the temperature, the higher the electrical conductivity would be. The electrical conductivity of water increases by 2-3% for an increase of 1 degree Celsius of water temperature. Many EC meters nowadays automatically standardize the readings to 25$^{\circ}$C.
While the electrical conductivity is a good indicator of the total salinity, it still does not provide any information about the ion composition in the water.
The same electrical conductivity values can be measured in low quality water (e.g. water rich with Sodium, Boron and Fluorides) as well as in high quality irrigation water (e.g. adequately fertilized water with appropriate nutrient concentrations and ratios).
http://www.smart-fertilizer.com/articles/electrical-conductivity
To start with, "water freezes faster when it starts out hot" is not terribly precise. There are lots of different experiments you could try, over a huge range of initial conditions, that could all give different results. Wikipedia quotes an article Hot Water Can Freeze Faster Than Cold by Jeng which reviews approaches to the problem up to 2006 and proposes a more precise definition of the problem:
There exists a set of initial parameters, and a pair of temperatures, such that given two bodies of water identical in these parameters, and differing only in initial uniform temperatures, the hot one will freeze sooner.
However, even that definition still has problems, which Jeng recognizes: first, there's the question of what "freeze" means (some ice forms, or the water freezes solid all the way through); second, the hypothesis is completely unfalsifiable. Even if you restrict the hypothesis to the range of conditions reasonably attainable in everyday life, to explain why the effect is so frequently noted anecdotally, there's literally an infinite number of possible experimental conditions to test, and you can always claim that the correct conditions just haven't been tested yet.
So, the fact that the internet is awash in a variety of different explanations makes perfect sense: there really are a bunch of different reasons why initially hotter water may freeze faster than initially colder water, depending on the precise situation and the definition of "freeze" that you use.
The paper you link to, O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox by Zhang et al., with results echoed by the HowStuffWorks video, attempts to solve the problem for a very specific sub-hypothesis. They eliminate the problem of defining freezing by treating freezing as a proxy for cooling in general, and directly measuring cooling rates instead. That experimental design implicitly eliminates one internet-provided explanation right off the bat: it can't possibly be supercooling, because whether the water supercools or solidifies when it gets to freezing temperature is an entirely different question from how quickly it cools to a temperature where it could freeze.
They also further constrain the problem by looking for explanations that cannot apply to any other liquid. After all, the Mpemba effect is about why hot water freezes faster; nobody is reporting anomalous freezing of, say, hot alcohol. That might just be because people freeze water a lot, and we don't tend to work with a lot of other exotic chemicals in day-to-day life, but choosing to focus on that restriction makes the problem more well-defined, and again implicitly rules out a lot of potential explanations ahead of time- i.e., it can't have anything to do with evaporation (because lots of liquids undergo evaporative cooling, and that's cheating anyway 'cause it changes the mass of the liquid under consideration) or conduction coupling to the freezer shelf (because that has nothing to do with the physical properties of the liquid, and everything to do with an uncontrolled experimental environment, as explained by John Rennie.
So, there really isn't just one answer to "why does hot water freeze faster than cold water", because the question is ill-posed. If you give someone a specific experimental set-up, then you can get a specific answer, and there are a lot of different answers for different set-ups. But, if you want to know "why does initially-hotter water cool faster through a range of lower temperatures than water that started out at those lower temperatures, while no other known liquid appears to behave this way" (thus contributing to it freezing first if it doesn't supercool), Zhang has your answer, and it's because of the weird interplay between water's intra- and inter-molecular bond energies. As far as I can tell, that paper has not yet been replicated, so you may consider it unconfirmed, but it's a pretty well-reasoned explanation for a very specific question, which is probably an influencing factor in a lot of other cooling-down-hot-water situations. There is a follow-up article, Mpemba Paradox Revisited -- Numerical Reinforcement, which provides additional simulation evidence for the bond-energy explanation, but it can't really be considered independent confirmation because it's by the same four authors.
Best Answer
The other answers are correct, but I think that you might benefit from a more "microscopic" view of what is happening here.
Whenever one substance (a solute) dissolves in another (a solvent), what happens on the molecular scale is that the solute molecules are surrounded by the solvent molecules.
What causes that to happen? As @Chris described, there are two principles at work - thermodynamics, and kinetics.
In plain terms, you could think of thermodynamics as an answer to the question "how much will dissolve if I wait for an infinite amount of time," whereas kinetics answers the question "how long do I have to wait before X amount dissolves." Both questions are not usually easy to answer on the macroscopic scale (our world), but they are both governed by two very easy to understand principles on the microscopic scale (the world of molecules): potential and kinetic energy.
Potential Energy
On the macroscopic scale, we typically only think about gravitational potential energy - the field responsible for the force of gravity. We are used to thinking about objects that are high above the earth's surface falling towards the earth when given the opportunity. If I show you a picture of a rock sitting on the surface of the earth:
And then ask "Where is the rock going to go?" you have a pretty good idea: it's going to go to the lowest point (we are including friction here).
On the microscopic scale, gravitational fields are extremely weak, but in their place we have electrostatic potential energy fields. These are similar in the sense that things try to move to get from high potential energy to lower potential energies, but with one key difference: you can have negative and positive charges, and when charges have the opposite sign they attract each other, and when they have the same sign, they repel each other.
Now, the details of how each individual molecule gets to have a particular charge are fairly complicated, but we can get away with understanding just one thing:
All molecules have some attractive potential energy between them, but the magnitude of that potential energy varies by a lot. For example, the force between the hydrogen atom on one water molecule ($H_2O$) and the oxygen atom on another water molecule is roughly 100 times stronger than the force between two oxygen molecules ($O_2$). This is because the charge difference on water molecules is much greater (about 100 times) than the charge difference on oxygen molecules.
What this means is we can always think of the potential energy between two atoms as looking something like this:
The "ghost" particle represents a stationary atom, and the line represents the potential energy "surface" that another atom would see. From this graph, hopefully you can see that the moving atom would tend to fall towards the stationary atom until it just touches it, at which point it would stop. Since all atoms have some attractive force between them, and only the magnitude varies, we can keep this picture in our minds and just change the depth of the potential energy "well" to make the forces stronger or weaker.
Kinetic Energy
Let's modify the first potential energy surface just a little bit:
Now if I ask "where is the rock going to go?," It's a little bit tougher to answer. The reason is that you can tell the rock is "trapped" in the first little valley. Intuitively, you probably can see that if it had some velocity, or some kinetic energy, it could escape the first valley and would wind up in the second. Thinking about it this way, you can also see that even in the first picture, it would need a little bit of kinetic energy to get moving. You can also see that if either rock has a lot of kinetic energy, it will actually go past the deeper valley and wind up somewhere past the right side of the image.
What we can take away from this is that potential energy surfaces tell use where things want (I use the term very loosely) to go, while kinetic energy tells us whether they are able to get there.
Let's look at another microscopic picture:
Now the atoms from before are at their lowest potential energy. In order for them to come apart, you will need to give them some kinetic energy.
How do we give atoms kinetic energy? By increasing the temperature. Temperature is directly related to kinetic energy - as the temperature goes up, so does the average kinetic energy of every atom and molecule in a system.
By now you might be able to guess how increasing the temperature of water helps it to clean more effectively, but let's look at some details to be sure.
Solubility
We can take the microscopic picture of potential and kinetic energies and extract two important guidelines from it:
Going back to the coffee cup question, all we need to do now is think about how these will play out with the particular molecules you are looking at.
Coffee is a mixture of lots of different stuff - oils, water-soluble compounds, burnt hydrocarbons (for an old coffee cup), etc. Each of these things has a different "stickiness." Oils are not very sticky at all - the attractive forces between them are fairly weak. Water-soluble compounds are very "sticky" - they attract each other strongly because they have large charges. Since water molecules also have large charges, this is what makes water-soluble compounds water-soluble - they stick to water easily. Burnt hydrocarbons are not very sticky, sort of like oils.
Since molecules with large charges tend to stick to water molecules, we call them hydrophilic - meaning that they "love" water. Molecules that don't have large charges are called hydrophobic - they "fear" water. Although the name suggests they are repelled by water, it's important to know that there aren't actually any repelling forces between water and hydrophobic compounds - it's just that water likes itself so much, the hydrophobic compounds are excluded and wind up sticking to each other.
Going back to the dirty coffee cup, when we add water and start scrubbing, a bunch of stuff happens:
Hydrophilic Compounds
Hydrophilic compounds dissolve quickly in water because they stick to water pretty well compared to how well they stick to each other and to the cup. In the case where they stick to each other or the cup better than water, the difference isn't huge, so it doesn't take much kinetic energy to get them into the water. So, warm water makes them dissolve more easily.
Hydrophobic Compounds
Hydrophobic compounds (oils, burnt stuff, most stains) don't stick to the water. They stick to each other a little bit (remember that the forces are much weaker compared to water since the charges are very small), but water sticks to itself so well that the oils don't have a chance to get between the water molecules. We can scrub them, which will provide enough energy to knock them loose and allow the water to carry them away, but if we were to increase the kinetic energy as well by increasing the water temperature, we could overcome both the weaker forces holding the hydrophobic compounds together, while simultaneously giving the water molecules more mobility so they can move apart and let the hydrophobic compounds in. And so, warmer water makes it easier to wash away hydrophobic compounds as well.
Macroscopic View
We can tie this back to the original thermodynamics vs. kinetics discussion. If you increase the temperature of the water, the answer to the question "How much will dissolve" is "more." (That was the thermodynamics part). The answer to "How long will it take" is "not as long" (kinetics).
And as @anna said, there are other things you can do to make it even easier. Soap for example, is made of long chain molecules with one charged end and one uncharged end. This means one end is hydrophilic, while the other end is hydrophobic. When you add soap to the picture, the hydrophilic end goes into the water while the hydrophobic end tries to surround the oils and burnt stuff. The net result is little "bubbles" (called micelles) made up of soap molecules surrounding hydrophobic molecules that are in turn surrounded by water.