What does a proxy do and how is it related to getting out of my network to the outside network?
Proxy means something like "a thing which acts on behalf of another thing". It's an english word, rather than an IT word. For example, voting by proxy is when you ask someone else to vote in your place, usually because you trust their decisions.
In the case of web proxies, it's a like a local webserver that can be asked for pages, but will get them from the real webserver and then give them to you. Likewise, in the case of a database server, a DB proxy will take a SQL query, send it to a remote database server, get the answer, and pass it back to you. In code, a proxy can be an instance (or object, as in object oriented programmming) which runs on the local machine and accepts commands, but actually passes those commands to a similar piece of code running on another computer.
Now, most of this doesn't sound too useful on its own. What makes it useful is that most proxies abstract the remote communication, or do useful intermediary things. For example, web proxies often get pages from a remote server, strip adverts or porn, save a copy, and THEN supply the modified webpage to the requester. If a second person wants the page, they get the saved copy. So that's pretty useful caching/modification work, and it's quite transparent to the user with their browser just asking for webpages as normal.
Likewise, a DNS proxy might accept requests for addresses on the internet, sending those requests to the internet as normal, but intercept requests for addresses inside the company and answer those itself using internal information.
In terms of "getting outside of your network"... it's the filtering thing. Sane companies lock down internet access, and force you to ask for pages through a web proxy. This proxy can then filter and monitor website requests, checking for porn, removing downloaded viruses, etc.
If you're talking about a proxy server for the web, it's a server that you make a request for a web page to, then it turns around and gets the page and hands it back to your machine.
Usually it caches the page in the process. Then if it's handling an office, for example, and you go to Microsoft.com, and three other people go to microsoft.com, the first request has the proxy go out and get the page, then the subsequent requests are pulled from cache, making them faster.
Proxies can also check to make sure the content isn't laced with malware, log your web browsing activity, or block you from going to porn sites or other types of sites that your employer doesn't want you using on company time.
Basically a proxy is a go-between agent between your intranet and the Internet.
Other proxies can be configured for services like FTP, etc. but the most common is for web browsing.