With GZIP on, phpBB2 utilizes ob_start("ob_gzhandler"), which will gzip (ie compress) the output buffer before sending the webpage contents to any web browser that can support this type of encoding. If the web browser does not support the compression, the webpage is sent as is.
This results in the pages loading faster, and saving on bandwidth as the webserver is sending less data out for each request as it is "zipped up" before being sent. There is a very minimal CPU increase to zip up the contents before being sent, but this is heavily outweighed by the bandwdith savings and speed improvements of loading each page.
With GZIP OFF, webpage contents are sent as is without any compression.
Hope this helps clear things up. For more info on this function, visit: