One way to conceptualize the average value of a function is to imagine it "melting" in order to level out and form a rectangle of equal area.
Another way to conceptualize average value of a function f(x) is to consider f(x) being a velocity. If so, then the integral of f(x) from a to be would be the displacement over that time interval. How else could something have that same displacement over the same time interval? By going the average velocity over that time... which would be a rectangle with a height of the average value of f(x) and a width of b-a.