I just got a new Macbook Pro. How do I transfer the license to the new computer?

You don't . . . you just reinstall it on your new computer.

When you launch it you will be given a registration screen which you can either complete or ignore.

If you choose to ignore it you will be reminded after every fifth launch.

There is no need or advantage to registering other than that it stops the "nag" screen.

Incidentally you may have problems installing it on a new computer as FCE 4 has been obsolete for over 3 years and is no longer supported by Apple.

Final Cut Pro X is the current editing app supported by Apple and is light years ahead of FCE.