Ok, umm proxy servers do three things generally.
1) All outgoing (from private network to net) traffic and all incoming ( net to private network) traffic passes through the proxy. This allows you to place certain restrictions on that traffic.
You can filter traffic by IP, Protocol, domain name, etc, etc. It is pretty flexible.
All traffic on the net appears to have come from the Proxy, not from the client ( i.e the packets have the Proxy's IP in the headers) This provides security for clients, as thier actual IP's remain unknown. This is known as NAT (Network Adress Transaltion)
2) Proxy servers perform caching services. Example: Popular websites that your users visit can be "cached" on the Proxy. When a client attempts to access the web page, instead of contacting the actual server hosting the page, the Proxy serves the client its Cached version. This reduces traffic on your internet connection and speeds up access time.
3) Some Proxies monitor internet traffic. This is generally how your boss finds out you have been looking at naughty things at work. I can pull up a user and get a list of every place he has been on the net.
I know there are tonnes of Proxy software available out there, some of it is free. Some has more functionality than others, but the premise is the same.
That's a simplified explanation. Basically it allows you to control, monitor and somewhat enhance the communications between a private and public network ( or private to private) you get the idea.