It looks like something is keeping the tray from closing all the way or the eject button is somehow being pressed when it closes. Open the tray and get a flash light to look inside to see if there is anything in there that shouldnt be (like maybe a disc that came loose from the tray and is still inside the xbox). If it looks all clear try holding down the eject button when putting the disc in. If that doesnt work try opening and closing it without a disc in. If it stays closed without a disc open and close it a couple times to be sure then try putting the disc in again. If that doesnt work try standing the xbox up on its side and putting the disc in.
The next step I would try is replacing the disc drive. This of course will void your warranty if you do it yourself I would suggest requesting repair from Microsoft if it is in warranty (http://support.xbox.com/en-US/xbox-360/repair/repair-process
If it is not in warranty try replacing the disc drive (This is pretty complicated. I have done it and honestly wouldnt suggest doing it yourself unless you are very familiar with disassembling and resembling electronics.) Here are instructions http://hardandsoftgaming.wordpress.com/2011/03/12/replacing-a-faulty-xbox-360-dvd-drive-firmware-flash-or-circuit-board-transplant/
If you are not comfortably with doing this yourself find a local repair shop.