If you follow harryb2448's directions, and if you have already established an account in the new version of OS X, the files will end up in a different folder than yours. What happens is Migration Assistant will create a new account so that it doesn't overwrite anything in YOUR account. That's safe, but to get the files to YOUR account you have to move them. The best way to do that is to go to that new account and login. It will use the same login and password as it had when the backup was made. Now select any files/folders you see and want and move them to Shared. Once you have them in shared, you can then log out, go to your own account and move them from shared to your folders, wherever you want them.
Now, if you don't want to go through all that, then attach the backup drive and then click on the TimeMachine icon on the Dock (across the bottom). What should happen is that the desktop is replaced by a Finder window in the middle of the TimeMachine desktop. Along the right side will be a column of dates that you can use to pick the date you want to restore from. When you pick a date, the Finder window will change, reflecting your system at the time of that backup. Now you can select files or Folders and then "restore" and they will be restored to YOUR account directly. It may take a while as the files will be coming over to the internal through a slower interface, but they will eventually be restored.
Finally, if you reinstall OS X and follow the process once it is installed, at some point it will ask you if you want to restore from a TM backup. At that point, before you have established an account, if you say "yes" then it will create a new user account from that TM backup, again using the same login name and password as when the backup was made. The files will then be restored to the same location as they came from and when it is all done you can simply log in and everything should be as it was, assuming all the applications you have will work with the new OSX.