BBCode stands for Bulletin Board Code. Bulletin Boards were the original file sharing forum type systems used on the early Internet before browsers came into being. Today, they are basically programs that run on the forum server to add functionality. You should leave it turned on.
I believe the only good reason to turn it off is when you don't trust the forum and want to eliminate any code from executing on another machine. Obviously that doesn't apply to our forum.
Frankly, I find that forums who offer this feature can be trusted anyway. Those that don't, can't!
Another way to look at it is that cut and paste is an OS feature that is supported by programs running on your computer. Cutting and pasting on the forum is a web feature running on the server not on your computer. So if you want to be able to run functionality like cut and paste on the server, you must let the server do that by enabling the code needed to do it.