This one started off as an expression that pissed me off- which, if you do not exist solely on sheets of cellulose and/or silicon, is not a very wholesome thing to do. But I digress.
The expression really means 'to leave' and usually, pretty quickly. (I know! I felt cheated too. How can you make like a tree and do something that a tree never does: to wit, leave?) A bit of googling, done a long time ago (which is why I cannot remember my sources) revealed that the expression was the result of a (rather sorry, in my opinion) pun on the phenomenon of leaving in deciduous trees, whereby they shed their leaves in "fall" (which is called so because leaves fall off trees in that season).
Therefore, you can make like a tree and leave. Though I still think it is kind of insensitive, given that trees stay 'rooted' to one place all the time. Wait till Arundhati Roy figures that one out...