why have LWP connect to a proxy?

To escape a firewalled network, mainly. They can also provide caching.

( For big web sites, sometimes you're actually connecting to a proxy. The proxy serves and caches the static content, while requests for dynamic content is forwarded to a server farm. )

When it connects to a proxy and fetches, lets say a file----does the proxy server actually do the requests and not the computer your on?

Your computer sends a request to the proxy. Your computer waits while the proxy repeats the request to the server. Finally, the proxy repeats to your computer the answer from the server.

would my access logs record the IP address from the proxy server or from the person's computer IP.

The proxy's. The user often isn't even on the internet, which is why he's using a proxy.