The Mean Value Theorem serves to show that for any given function that is continuous on the closed interval and differentiable on the open interval , there exists at least one point in the open interval such that the instantaneous rate of change (the derivative) at that point is equal to the average rate of change over the entire interval.
You can think of this saying that the rate of change has to “cross the boundary” at some point to get from the starting slope to the ending slope.