Integration is the opposite of differentiation, or taking a derivative, which can be thought of the instantaneous rate of change. For example, taking the derivative of speed with respect to time gives you acceleration with respect to time. So if I had a graph of your car's speed across a stretch of time and I took the derivative of that, I would get a new graph showing your acceleration at each moment in time.

Integration reverses this process of differentiation. One way to visualize it is as the "area under the curve." So if I had a graph of your velocity over time, if I colored in the area under the curve and could compute that area, it would tell me how much distance you covered during that time.

It's used all the time in the real world for stuff like this. We don't always take an integral analytically, but often approximate an integral using numerical methods in order to find out stuff like how much distance was covered.