I have linux mint 15 mate. I have installed xbmc 12.2 but when I start it up it goes to the boot logo of xbmc and I can see the mouse but it crashes short after that. I am new to Linux and xbmc
Create a debug log if you want help, but I dare say it is likely that you don't have a proper accelerated video driver for your GPU running.
What is your GPU? (lspci is the appropriate utility)
What driver is X running? (take a look in /var/log/Xorg.0.log)
If you want to paste a log use pastebin. There is an excellenet utility called pastebinit which you can install. Then it is simply a matter of running
lspci|pastebinit or
pastebinit /var/log/Xorg.0.log
Note the link reported back and paste it in here.
Your logs confirm your video card and/or drivers aren't up to snuff for xbmc. Beyond that I am more of an ati/amd avoider than expert.