What is the point of burning Xbox games to your hard drive?

Exactly, Microsoft wasn't quite going to make it so that people could rent or borrow games, install them, and keep them for free, which is why it still requires the disk to be in the system. Installing game disks still has bonuses though: reduced wear and tear on the system and disk. reduced noise and heat production; some games experience reduced glitches and loading times; and if your disk gets damaged, you can continue to play the game as long as your system can read the beginning of the disk and start the installed data.

The purpose is to enhance game performance. When it is on your hard drive it will decrease things like load times because the console isn't having to process the information from the disc into the console, it is already on the console. When the game is installed when you put in the disc the disc is only there to get past the security check and then is unneeded other than to just be there (if that makes sense.) This also means that if you get a really bad scratch on your disc (such as the perfect circle) more often than not you will be able to clear the security check and play your game as normal. You have to have the disc in for the obvious reason you mentioned that it stops people installing borrowed games.