The Xbox has an option that allows you to install a game to the hard drive. When you do this it will make the game load speeds much faster. When the game is installed to the hard drive, I try to load it up, but it still requires the disc. Why is this?
As part of security and to stop you screwing people over, you must keep the disc in order to play the game. Think of it this way, if you have a big 2tb hard drive, you could rent out 50 games, rip them to the HDD and now you have 50 games that you never paid for. It would also mean you can trade in games without actually getting rid of the game. This would be a serious nightmare for Microsoft and any developers involved.
Installing a game to the hard drive will greatly increase the load speeds for some games and wil reduce some wear and tear on your disc drive, but it will not allow you to play games without the disc.