It's a simple concept.
A proxy is another computer that you use in between your computer and the internet.
There is an application running on this computer (the proxy server). Once you configure your browser to forward all requests to the proxy server, this is what happens:
Say you type in
www.google.com in your browser.
The browser makes a TCP packet with an HTTP GET
www.google.com request (with sender address as your IP address) and sends it to the proxy server.
The proxy application running on the server looks the packet and sees that it is addressed to google. It forwards the packet to google.com, changing the sender address in the TCP packet to its own.
[NOTE: This is where the anonimity comes in.. Now google doesn't know that it i you who is using their service, they only see the proxy server using it.]
Google gets this packet from the proxy and sends the contents of its website in another packet (well actually a series of packets) to the proxy. The proxy accepts this packet and forwards it back to you.
Proxies may or may not be safe. If the proxy application has a logging feature, all your keyboard activity, your usage logs including passwords, credit card information, EVERYTHING can be stored in the proxy computer.
