An anonymous proxy server hides an IP address by making requests for data on another computer's behalf.
If you are browsing the web through an anonymous proxy server, your computer tells the proxy server what website you want to view. The proxy then connects to the website for you. The website uses the IP adddress of the proxy server to send back the requested data.

The proxy server then sends the data you requested (the website) back to you. Not all proxy servers are anonymous. Also, for the anonymous proxy server to work as intended, it needs to be configured properly.