|MadSci Network: Physics|
That's a good question. The thing about elementary particles is that they are supposed to be elementary, which is to say that they are not (currently) thought to have any underlying structure. So things like their charge, mass, etc, would be basic properties of the particles themselves.
In the case of electric charge, for example, physicists define charge simply as the particle's property that determines how it is affected by electromagnetic fields (or more accurately, its coupling to photons).
Your question, though, is why elementary particles have charge. We don't have an answer for that yet. The best we've been able to do up to now is to construct a pretty good mathematical description of how particles and fields interact with each other, and plug in the measured properties of the particles we see.
There is a hope that someday we will reach a deeper understanding of the fundamental particles and forces, and how they are inter-related (as we think they may be). At that stage, perhaps we will gain a better understanding of why all of the different particles we observe ended up with their respective charges, masses, and so forth. But we'll have to wait and see.
I hope this was helpful. Keep asking questions!
Try the links in the MadSci Library for more information on Physics.