Very confused on integration yielding area under curve

In my calc class, we've been on integration for a while, and all the time that we have been learning more advanced integration techniques, I'm still dumbfounded on why finding and antiderivative will yield the area under a curve. I easily understand how to find an antiderivative, and understand why F(a) - F(b) gives the area between the two points, but am absolutely lost when it comes to why the antiderivative of f (F) will yield area when the points are plugged in. I wouldn't waste anyone here's time if I understood the proofs I've seen, trust me. Could someone weigh in with a simple enough proof, or just help me understand the concept? Thanks!

So you know the f(a) is the value of the function at $x=a$ and f(b) is the value of the function at $x=b$. Think of this as drawing a really thin line from the x-axis to the value of the function.
In order to get the area under the curve we need to draw infinitely thin lines at all the points on the interval [a,b]. That is we need to draw infinitely thin lines from the x-axis to the functions value over the interval [a,b]. We will define this as f(F). This will eventually span a shape under the graph from the x axis over [a,b] to the points f(x) $a\leqslant x \leqslant b$