During the cold war, Russia/China and USA fought proxy wars, for example in Korea or Vietnam (China supporting the communist Viets, USA fighting with the whatever, liberal Viets or smthg). USA and China didn't fight directly, but used smaller countries as proxies to avoid direct confrontation for a number of reasons such as USA kicking China's ass.
Now your computer is China, the proxy is the communist Viets, and the USA is the internet.
1) You don't want to get blacklisted (get your ass kicked) or sandboxed, so you use proxy servers to send your spammy requests.
2) You want to create multiple accounts on say Facebook but FB only accepts like 5 accounts per IP so you want to simulate multiple IPs (ie. looks like you are from new location each time)
3) You want to hide your IP for some obscure reason.. although this is not exactly bulletproof.
That's the main reasons why you want to use proxies, now don't ask me how they work technically because I have no idea and I'm not interested, as long as they are working and I know how to use them, I don't need to know anymore.