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 I would swap who I'm dating with who the last person who teased me is dating."
Suddenly, Biff's girlfriend (his girlfriend in the new reality, who was a geeky girl named Sophie Stewart) swapped with Randy Goodman's girlfriend (who was the cheerleader Jessica Davis). But to everyone else who was not associated with the wish, Biff was supposed to be dating Jessica, and Randy was supposed to be dating Sophie.