Basically, the router connects to the web, and is assigned a temporary identity by the service it connects to. All computers that go out through the router look like they are "that same identity," and hence are "disguised" so far as the outside world is concerned. The identity of an individual computer going through the router cannot be determined from the outside, and this is what's illegal as the new state laws are written.

This probably oversimplifies things, but I think it's the gist of it.

John