Physical Intuition

Divergence (div) is the total amount of flux going into or out of a point.  You can think of it as the rate of expansion (positive div) or contraction (negative div) of a field at a point.  Imagine that you weren't insane but could actually talk to point in a field and ask it what it saw. Remember that we take flux to be positive going outward.

If a point saw all bunch of flux coming towards it, he would scream and say that everything was closer.  He would have negative divergence.

If a point saw all the flux leaving him, he would sniff his armpits and then say everything was going away.  He would have a positive divergence.

Think of divergence. as the net flux per unit volume. So, when you have flux exiting a given volume, we have positive divergence., and the vice-versa for negative. It is pretty easy to figure out: the very word "diverge" means to move away from.  This concept seems pretty simple, and it is, but it is useful in some cool theorems.