Dropping the mic, according to the Urban Dictionary, involves "Calling someone out so hard that you just walk away indisputably victorious." Those employing this communication strategy often actually drop a mic or pretend to do so.
In communication terms, dropping the mic represents a form of discursive closure, a concept that refers to a number of strategies designed to end discussions and cut off any further response from others. In short, dropping the mic is a way to end the conversation when the communicator is at an advantage. Traditional forms of discursive closure include naturalization, which is performed when a person says, "That's the way it is," and topical avoidance, which prevents some topics from being discussed.
To see how deGrasse Tyson produces discursive closure by dropping the mic, watch the following video of his appearance on The Nightly Show in January:
As you can see, deGrasse Tyson uses the mic drop to end the argument he was having with B.o.B about the shape of Earth. The action communicates that B.o.B's opinions will no longer be entertained by deGrasse Tyson. He's also using the communication strategy in an attempt to head off science misinformation in general. Settled scientific fact is left in a position of power, and those who might seek to challenge it are dismissed.
And that's the way it is.