What does "Verify Integrity of Game Cache" do?

It compares the game files you have on installed your computer with those of of the latest version on the steam servers. If its different in any way it replaces or adds on any files.

If I ever have a problem with a game on steam this always the first thing i do.

The Steam Client conducts a checksum of the *.manifest files for the game. If your local checksum mismatches the checksum on the Steam Servers, the client will download the files that are "corrupted".

In other words, say your game is several word files in a folder. It will go through each file and count all the a's, b's, ..., z's in the files. If the count is mismatched with the Server record, you will be triggered to redownload the errant file(s).