A proxy simply takes your browser (or whatever software you use with it) and reroutes the traffic through another location. This causes you to appear to come from that location instead of where you really are.
If you need to do keyword research for Toronto for a customer, using a proxy in Toronto will allow you to see the same results as someone searching from Toronto, making your work much more accurate.
As you travel around the Internet, your IP address is what you are "known" by. When you build a website and load it to your server your IP address is recorded. Anywhere you go, your ip address is tracked for statistical purposes.
Suppose you built your website, then went out for a week leaving really high quality links where you could. Each of these links will have come from the same IP address, and be a clear indicator that you are making your own links. Which Google REALLY hates now.
If a bunch of web 2.0 properties point to your money site, and they all were built and maintained from the same IP address it is another nail in your coffin.
If you are doing a lot of research and using automated tools, Google (and many servers) will actually disable your internet connection with them for an hour or so. If you did this with a list of proxies to rotate thru, the connections would be coming to Google a few at a time from all over the place, not from your connection, so it will not look like you are doing it, so none of them get turned off.
If you were to post a bunch of links from proxies, it looks like other people are leaving comments about your site, not you. If you were to blast out a bunch of links using proxy it will make them appear to come from all over the palace instead of you.
If you were to , ahem, get access to, say, a paid membership site from a link you somehow found out about, and you were not really a paying customer, but really wanted to visit the site, you know just to look around, maybe download a file for a test or something, but you didn't want someone to find out you did and tell you that you were on their site without being a member, then you might decide to use a proxy instead of your own browser IP, which would keep your real IP address from being recorded.
Not that I condone such behavior, you understand.
A proxy works for the piece of software you set it up on, not everything on your computer.
If you get a VPN (virtual private network), it does the same thing for everything your computer does, instead of being software specific.