[Polygon]

The turbo connects to the exhaust at one point, normally the manifold. The exhaust gas spins a turbine which is connected to another (compressor) turbine via a shaft. This turbine compresses the intake charge and sends it through the intake plumbing and forces more air into the cylinders. This allows you to add more fuel. Thus, more power.

That is the simplest explanation I could think of. For something more detailed check How stuff Works as Sleepr awd suggested.

Most of the time this would only be illegal to drive on public roads is if it wasn't able to pass emissions. However, California has come up with some bullshit excuse to nail you for modifying your car. I'm sure this is to curb street racing.