What the fuck does reversing the derivative operation have to do with summing up infinitely thin(enough) rectangles under a curve?

I remember thinking the exact same thing when I learned integration, here's what made it eventually click for me (though this is not a formal proof by any means)

Let the area under the graph of y = f(x) from 0 to x be A(x). If you extend x by a tiny amount h, that's like adding one of those tiny rectangles to the sum - in this case that rectangle will be h \* f(x). When h is small, you expect this to be close to the actual difference of A(x) and A(x + h). In other words:

A(x + h) \~= A(x) + hf(x)

Then by rearranging:

f(x) \~= (A(x + h) - A(x))/h

When h tends to 0, this gives the exact defintion of the derivative, showing that the function of the area under a graph of a function differentiates to give the function itself. A stronger argument would show that the approximation tends to an equation as h tends to 0.
If you walk in a straight line at various speeds, the distance you travel can be determined by a weird kind of sum of how fast you were walking and for how long you were walking that fast. Does that make sense? If you walk 5mph for an hour you've walked 5 miles.

That is intuitively equivalent to this idea. You've summed up "rates of change in position" i.e. "the derivative of position" i.e. "how fast I'm walking along the straight line" according to how long you walked that fast, and you got a distance.

"The sum of small changes times how long we were changing that fast tells us how far we've moved" rather "how far we've moved is the sum of small changes times how long we were changing that fast"- this is the fundamental theorem of calculus.

Edit:

This is also the intuition behind the Stokes theorem, it's special case Green's Theorem, and also the generalized Stokes theorem (of which the Fundamental theorem of calculus is actually just a special case).
Suppose you didn't know how much money you had in your bank account at the moment, but you did know how much you had a month ago, and you knew what you added and subtract each day since then?

If you added up all those changes over time, then you could add that to what you had a month ago and you would know what you have now.

If you multiply the slope by a tiny x-slice you get a small change in y value. If you multiply a y value by a tiny x-slice you get a small change in area.

Integrate y-values, get an area.

Integrate slopes, get a y-value.

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