A bad pun, based on the fact that:
'leaf' as an intransitive verb used for trees (to put forth leaves, to bud) sounds similar to 'leave' (to go away).
So it's basically telling someone, 'Be like a tree - go away.'
People here say it at parties when they're bored and want to go: 'Let's make like a tree and leave.'
Other examples would be:
'Let's make like a dog and flea' - dogs have fleas, flea sounds like flee (to run away)
'Let's make like a shepherd and get the flock out of here' - flock (of sheep) sounds like f***
I could go on , but you get the idea...:)