To cut, as you already know, is generally 'to slice with a sharp object'; if you "cut your own arm", you take a knife and make an incision somewhere on your arm.
To cut out is probably, in this case, 'to free from entrapment by cutting the trapping material'. If you "cut out your own arm", then somehow one arm is trapped, perhaps under a fallen log, and you use your other arm to cut the log until you can get your first arm free. (This usage seems very unlikely.)
The most common use of cut out is 'to shape an item by cutting a larger element'; for instance, if I have a piece of paper, I could 'cut out a paper doll'.
To cut off is to sever or detach; if you "cut off your own arm", then either you no longer have an arm, or you need major surgery to get it re-attached.