How much of a flow actually gets through a surface — and why you have to decide which side counts as out before the question means anything.
Warm patches are places the field leaves through; cool patches are places it enters through. Flux is their tug-of-war.
Tilt the surface, then flip its orientation and watch the sign go.
Hold a hoop up in a river. Flux is how much water per second actually goes through it. Turn the hoop edge-on to the current and the answer drops to nothing — not because the river stopped, but because nothing is crossing the hoop any more.
So flux can't just be about how strong the field is. At each little patch of surface only the component of F along the normal counts, which is the dot product F · n; the part sliding along the surface contributes nothing. Add that up patch by patch, weighting each by its area, and you have ∬ F · n dS.
Which brings up the awkward question the button in the corner asks. A surface has two sides, and the normal could point either way. Nothing in the surface itself decides which — you have to choose, and until you do, "the flux through this surface" is not a well-defined number. Flip the choice and every patch changes colour and the total changes sign.
That's not a flaw, it's the point. For a closed surface everyone agrees to point the normal outward, which is what makes "flux out of this region" unambiguous — and that convention is what the divergence theorem is written in.