Shrink a closed surface toward a point and the flux per unit volume settles onto a single number: the divergence there.
Shrink the radius and watch the last two numbers meet. On the inverse square field, drag the centre until it swallows the origin.
Shrink the surface and watch the ratio converge.
Wrap a closed surface around a chunk of space and measure the net flow out of it. If more leaves than enters, something inside must be producing the stuff. The divergence theorem makes that accounting exact: the flux out equals the total production inside.
"Production per unit volume, right here" is what div F means, and the two bottom numbers are the definition being tested. Divide the flux by the volume it came from and you get the average production in that ball. Shrink the ball and the average has nowhere to hide — it converges on the value at the single point at the centre.
That's the whole trick. A quantity defined by a big global integral turns out to be an integral of something purely local, and the local thing is a derivative you can compute without ever drawing a surface.
Then there's the inverse square field, which is the one worth playing with. Its divergence is zero at every point where it is defined, so any surface you draw has zero flux — right up until you slide the centre so the sphere swallows the origin, at which point the flux jumps to 4π and stays there no matter how you resize it. The theorem isn't broken; the field simply isn't defined at the origin, and all the production in the universe is concentrated at that one missing point. That is Gauss's law, and it is why a point charge has a flux and not a divergence.