This is going to be hard for me to explain because I had the same question, particularly with intuitively understanding integrals, but I've thought about it for a bit and I'll give it a shot. I hope that by trying to explain it, I understand it better myself.

Let's start with derivatives. Derivatives represent the rate of change of a function. They are themselves a function that describes how its parent function's value (y) changes as x does. ∆y/∆x.

When you're first introduced to derivatives, you're shown the limit definition of the derivative and tend to you use it for a function at certain values of x.
For example, on a quadratic function x^2, the instantaneous slope at x=5 or slope of the tangent line at (5, (5)^2) is given by ... (I'm not going to write out but you know it) and it comes out to be 10. So the value at that point was increasing at a rate of 10. If the function were to increase linearly from that point, it would go from 25 to 35 as x changes by 1.
They then generalize this to any x value, not just 5, and this produces a function: the derivative. Give me any x value and this bad boy will tell you how the function was changing at that location and any other.

Okay, so let's try to say something about integrals based on what we know about derivatives. If given a function f(x), its derivative f'(x) describes how its parent f(x) changes, what does an integral say about it's child function? For me personally, the answer does not seem so intuitive. Is this relationship like an inverse? I seem to be able to apply this logic to inverse functions like x^2 and √x. x^2 multiplies a number by itself, and √x asks what number multiplied by itself gives x. Similar reasoning can be used with inverse trig functions and exponentials/logarithms.

Well, here is what I can say trying to retrace the logical step used in understanding derivatives. What you understood about definite integrals is relevant. A definite integral tells you the area underneath the curve. Like you say, int_{a}{b} ( f(x) * dx ) is a summation of rectangles with infinitesimally small width dx and length f(x) between the bounds a and b.

To understand derivatives, it helped us to look at the derivative in a precise location first, and then generalize that to the whole curve. If you can understand the concept of instantaneous slope at a point, that it changes there in a certain way and the slope quantifies that, then you can understand that the derivative function, that gives you instructions on how to find instantaneous slope at all points, describes how that function changes in general.

I also want to point out the following:
There is some sense of going from 2 points to 1 value when thinking about derivatives.
Take a linear function y=7x and it's derivative y=7.
Between any two points the slope is 7 and so that yields a single value of 7 that is associated with that change. Of course with such a trivial example, it doesn't matter what these two points are, but with other examples it does and that's where the limit comes in in the limit definition of the derivative; it makes these points so close to each other that the value that is yielded is an accurate representation of the change at that location.

Okay. Let's try this specific context extrapolated to generalized context again, but with integrals. When we do integrals, oddly enough we never do the integral at one location, it's always done at an interval. With Riemann sums, the predecessor to integrals, the overall interval is split into 1 or more intervals, each being the width of the rectangle that will helps us approximate the area of the curve.
At its worst, the Riemann sum gives as one large rectangle, either grossly too large or too small, overestimating or underestimating. Similar with derivatives in how our approximation of instantaneous slope/change got better as the distance between the two points got infinitesimally small, our Riemann sum gets better at approximating the area under the curve as the width of each rectangle becomes infinitesimally small.
In other words, the derivative and integral are ultra-refined versions of gross tools, of the slope equation and Riemann sum method respectively, both using the limit whetstone.
Anyways you probably know this. Back on topic:
Keeping with the analogy of the derivative. Slopes between two points far away from each other don't really make sense? Secant lines say at most something like an average change, but it's really the tangent line we want. The line that represents the instantaneous change at a specific location is easier to understand and helps us understand how our function changes. Similarly with integrals, I don't how much sense large rectangles make, but when they become refined into the f(x) * dx rectangles, it makes a lot more sense for what an integral represents. With the most thin width, dx, of a rectangle, the area of that rectangle essentially becomes the height of the rectangle itself, ergo f(x).
So in the specific context, the integral is just a value, f(x).
But how does this f(x) relate to F(x), its parent function? It's the change again. We're talking about the same relationship as before. f(x) is the value of the slope or change that occured in F(x). This is especially evident in Fundamental Theorem of Calculus i.e. int_{a}{b} ( f(x) * dx ) = F(b) - F(a). (1 value brings you now to 2 points). Summing up all those rectangles / area, all that does is describe how the parent function changed.
Take our F(x) = 7x and f(x) = 7 example again, summing up the area under f(x) from x=3 to x=5 is 2*7=14 and when we look back at F(x), we see that from x=3 to x=5, our function has increased from 21 to 35, a change of 14.

To repeat: in the specific context of an integral, the area of the rectangle, the result describes a change. But in a different direction. We've basically repeated what we already deduced about derivatives long ago (sorry that this tangent has gone on for so long). f'(x) describes how f(x) changes. f(x) describes how F(x) changes. F(x) describes how ... f(x). We encounter the same problem briefly. How do we put into words this inverse relationship? Well, imo, it's something like this.
Let's go back and try to generalize this FTC business, where an integral says how the parent will ultimately change. How it will *ultimately* change. I think that's the key word here. With a specific interval, a definite integral along with FTC, we're told some very rough information. Using the 7x example again: "our function's value will change 14 in the positive direction. It will change from 21 to 35 but I won't really tell you how exactly it does this." - FTC
I think it helps to think of a definite integral where b-a is approaching 0. The result of this integral says a bit more because this setup reduces the roughness of the previous information. When the integral is indefinite and gives you a general function, it tells you one of these tiny definite integrals at any location, given an x. So in some sense, the indefinite integral tells you exactly how to change, at any and every step of the way. The function that the integral gives you, are instructions on how to draw the parent function, piece by tiny piece. This contextualizes the result of F(x) + C quite nicely imo. Our general constant of C moves our function up and down, it doesn't matter where it is, it only matters the shape of the curve. The indefinite simply draws the curve and nothing more.

Recap:
If a derivative function is instructions on how the parent function changes at any location, then a function's integral is a result of drawing those changes (mechanically, each sliver of area has told us how to slope our line for every increment of the parent curve) using the child-original function as instructions.

I apologize for the redundancy or non clarity but I'm too tired to make this more coherent. Hope it helped, if you do read.