Biff grabbed the note. Not because he wanted to read it, but because his curiosity of the "magic note" outweighed his common sense.
And when the note got close enough to him, Biff felt compelled to read it. "I wish that the last person who teased me would become whatever they called me."
Immediately after Biff made that wish, womanizer and jock Randy Goodman (the last person who teased Biff) suddenly felt dizzy. Soon, he would become a "dickweed", which was what he called Biff when he bullied him.