Files that are compressed make downloading them easier. But what do I do once I've downloaded my compressed .tar.gz file?

You have some choices when it comes to decompressing these files.
If you're on a Linux machine and you want to decompress a file in the current working directory, you can execute this command

$ tar -xvz drupal-6.x-dev.tar.gz


Or if you're on Windows you can download 7zip here: http://www.7-zip.org/
Once installed, you can navigate to the compressed file and right-click to "Extract Here"