Well, far from being an expert, I'll give it a stab:
In very simplistic terms, the turbo has two wheels (fans), connected bt a shaft. Exhaust gasses spin one wheel, which in turn spins the other, which takes atmospheric air, compresses it and forces it into the intake manifold.
Remember, in a Normally Aspirated (N/A) engine, the combustion chambers fill by the vaccuum caused by the piston moving down the chamber.
The compression of the air causes it to get very hot. The intercooler cools the air, making it denser and allowing more ignition timing (hot air causes detonation, all else being equal).
Hope that helps and that I'm talking crap. There are some VERY turbo savvy members on this board, hopefully theyu will chime in.
Tim