To be just a little more succinct than Jurph
, the mean value theorem says (in our speeding metaphor) "You can't average a speed without going that speed for at least a moment".
Remember rise over run? That's exactly what the right side of the function is. f(a) - f(b) is the rise of the function (the difference between the top and bottom) and a - b is the run (the length of the interval of the function. So that's the slope of the straight line connecting f(a) and f(b). However, our function f(x) might have lots of curves; it can pretty much go all over the place. If we take a point on the curve of f(x), and find the slope of the curve at that point, we call that f'(x) (we say "f prime of x"). So for a point c that is between a and b, we can find a slope for that point, and we call it f'(c). In a way
In our example, a, b, and c are times - the time we lease the first toll booth, the time we arrive at the second, and some time inbetween, and f(x) is the distance we have gon at that time. The right side of the equation above is average speed, and f'(x) turns out to be the speed we're going at at that moment. So f'(c) = (f(a) - f(b))/(a - b) says that we must have been going our average speed at some point during our trip.
(P.S.: do they really do that with toll booths? That's kind of evil...)