According to Lenz's law, polarity of induced emf will be such that it produces the current which opposes the change in magnetic flux which produced it. It directly follows from the law of conservation of energy.
Let's say you have a circular loop of conductor, if you bring the north pole of the magnet towards the loop, current induced in the loop will be anti-clockwise when viewed from the side of the magnet. Remember if current is in the anti-clockwise direction it acts same as the north pole. So, north pole-north pole repel each other. Thus, induced emf is opposing the change in magnetic flux. Therefore, work has to be done in order change the magnetic flux linked with the coil. This work done will be converted into electrical energy.
When you take the magnet away, when viewed from the side of magnet, current in the loop will be clockwise (acts as south pole) whereas magnet side near the loop will be south pole. So, even here south pole-south pole repel each other. Thus, induced emf is opposing the change in magnetic flux. Therefore, work has to be done even here in order to change the magnetic flux linked with the coil. As said above, this work done will be converted into electrical energy.
Thus, in the above two cases energy is conserved.
Let's say that you bring the north pole of the magnet towards the same circular loop, now let current induced be clock-wise (acts as south pole) when viewed from the magnet side. North pole- South pole attract each other, so no work is needed to change the flux linked with the coil. But when flux linked with the coil changes, there is emf induced. Here, electrical energy is produced without any work being done. It violates the law of conservation of energy. Similarly you can consider the other case.
Therefore, from Lenz's law it follows that electrical energy is produced by the expense of mechanical energy. For this to happen, induced emf should oppose change in magnetic flux.
You asked to explain with out using law of conservation of energy, but I felt the above explanation would explain your questions better.
As the magnet approaches the solenoid, a current is induced. The current generates a magnetic field. The field repels the magnet, slowing it's approach. The amplitude of the oscillations diminish.
If there was no resistance, this would work in reverse as the magnet receded from the solenoid. The magnetic field would accelerate the magnet. The magnet would induce a current in the other direction, reducing the current to 0. This would reduce the field of the solenoid to 0. The amplitude would not diminish.
Best Answer
You can actually arrange a simpler experiment for this. Suppose that Lenz's law were reversed and induced currents reinforced the change of the magnetic flux. Now take a single loop of wire, and suppose that we produce a small current in it (with a battery or a magnet - it doesn't matter). An increasing magnetic flux is then created through the loop.
This changing flux will now induce a current in the wire loop. You'll see by the right hand rule and the modified Lenz's law that the induced current goes in the same direction as the existing current. So the induced current reinforces the existing current - the total current increases and so does the magnetic flux, which induces a further increase in the current etc... With a simple loop of wire you could power a city.
In the actual case the induced current resists the increase in mangetic flux and opposes the current already present in the wire, slowing and eventually stopping the growth in the current. This is the operating principle of an inductor.