Intuitive

Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative divergence). If you measure flux in bananas (and c’mon, who doesn’t?), a positive divergence means your location is a source of bananas. You’ve hit the Donkey Kong jackpot.

Vector Calculus: Understanding Divergence by Kalid Azad

Concrete

• Vector Calculus: Understanding Divergence by Kalid Azad
• Another great discussion of the divergence in vector calculus can be found in Section 3 of Vol. 2 of Feynman's Lectures on Physics, which is available here

Abstract

Why is it interesting?

The divergence is a crucial tool, for example, in electrodynamics. This can be seen immediately because the fundamental equations of electrodynamics (called Maxwell equations) contain the divergence of the electric and magnetic fields.