When physicists say that a particle has electric charge, they mean that it is either a source or sink for electric fields, and that such a particle experiences a force when an electric field is applied to them.
In a sense, a single pair of charged particles are a battery, if you arrange them correctly and can figure out how to get them to do useful work for you. It is the tendency for charged particles to move in an electric field that lets us extract work from them.
A typical electronic device uses moving electrons to generate magnetic fields (moving electrons cause currents, and currents generate magnetic fields) and these magnetic fields can move magnets, causing a motor to turn. What is happening at a fundamental level is that an electric field is being applied (via the potential across the battery) that is causing those electrons to move.
If I wanted a magnetic field to be generated, I could get one from a single pair of charges, say, two protons placed next to one another. The protons will repel (like charges repel) and fly away from each other. These moving protons create a current (moving charge) which creates a magnetic field.
Your author is right when he says that charges attract or repel other charges. To help connect it to more familiar concepts, consider this: The negative end of your battery terminal attracts electrons and the positive end repels them. (The signs of battery terminals are actually opposite the conventional usage of positive and negative when referring to elementary charges. As a physicist, I blame electrical engineers.) The repelled and attracted electrons start moving, and these moving electrons can be used to do work.
Get together a collection of charges. As many different ways to generate a charge as you can think of. Go ahead and invite your friends so they can think of some more. (As a practical matter you make static charges just before you use them, but still...)
Now, test them pair wise to see if they attract or repel one-another. Keep careful records.
Find the largest set that are all mutually attractive and the largest set that are all mutually repulsive.
You'll find that the attractive set has exactly two members (though you can make many different examples of this set) and the repulsive set consists of half (either half!) the charges you've created.
Ponder that for a while. It also gives you the answer to how like charges respond to one another (though you can get that directly by preparing two similar charges).
Best Answer
On an atomic level, what you have is charged particles that attract and repel each other based on their electrical charges, through the very same electrostatic interactions you just learned about.
The electromagnetic interaction is fundamental, and it cannot be explained in terms of anything else; you can pretend to explain it through other means but that is ultimately creating circular reasoning. There's a reason that electromagnetism is a fundamental interaction - it just something that happens.
As usual, though, (if you swap charges for magnets), Feynman explains it best, in the classic Feynman interview on magnets and 'why?' questions, so I'm going to shamelessly steal from the transcript: