It allows you to have more control over the phone. You can install apps that require root access, like the Tether for free mods, Titanium Backup, etc. It also allows you to install custom ROM's. And it lets you tweak system files to do things like make the home button stop waking the phone up. These are just examples.

I root my phone mostly for the free tether ability and other minor tweaks. I leave the ROM stock as I like Touch Wiz. My wife's S3 has never been rooted, and that fits her non-techie needs.

 

Red through this post for explanations on rooting and what it does.

http://androidforums.com/faqs/443432-what-root-being-rooted-mean.html