# cryptsetup luksOpen /dev/sdX mydisk
(You will be prompted for password)
# mount /dev/mapper/mydisk /mnt
Then files will be accessible under /mnt. Typically a GUI might do this for you, see gnome-disks, maybe nautilus too. There is no reason why you couldn't access the files really, everything should work normally, given you put in the password correctly and you have the correct drivers (only have to worry about that if you compile your own custom kernel).