Why are slopes and areas two sides of the same idea?

Slopes and areas are two sides of one idea because adding up and measuring change undo each other. The fundamental theorem of calculus links differentiating a function, calculating its slopes, with integrating a function, calculating the area under its graph. Roughly speaking, the two operations can be thought of as inverses of each other.

Picture a car. You can see the velocity on the speedometer but cannot look out to see your location. Each second you can work out how far the car has gone from the speed, and add it up. That adding up is integration, and it turns the speed into the distance. Going the other way, if you knew the position at every moment, the steepness of that position graph would give the speed. That is differentiation. The theorem says the two trips return you to where you started.

Here is why. Let the area under a curve be counted from a starting point up to x. Nudge x forward a little, by h. The area gains a thin strip, which is about as big as a rectangle of width h and height f(x), the height of the curve. So, for a continuous curve, the area grows at a rate set by the height of the curve. The derivative of the area is the original function, which is the essence of the theorem.

Before this was discovered, it was not recognized that the two operations were related. Ancient Greek mathematicians knew how to compute area, and scholars had studied motion long before. The historical relevance of the theorem is the realization that the two seemingly distinct operations, areas and gradients, are actually closely related. It also saves effort: a definite integral can be found from the change in an antiderivative between the two ends, without adding up slices.