TITLE: Intuitive explanation of the Burnside Lemma
QUESTION [13 upvotes]: The Burnside Lemma looks like it should have an intuitive explanation. Does anyone have one?

REPLY [3 votes]: You can quickly reduce to the case of a transitive action, in which case we just want to explain why the total number of times that something gets fixed is exactly the size of the group.  But in this case everything is symmetric at all points in the (unique) orbit.  So to count all the times something gets fixed, we can just count how many times a particular x gets fixed, and multiply by the size of the orbit.  Now we've reduced to the fact that the size of an orbit is the index of the stabilizer.