When first learning calculus, you are taught that the first derivative taken at a single point is equal to the slope of the equation at that same point.
Here is a visual interpretation of the equation that can be used to convince you that this is true.
Dragging the yellow point down the function closer and closer to the red point demonstrates the limit being taken as the distance h approaches 0. When h finally reaches 0, the slope of that function is the green "m" value.
NOTE:
RISE is the difference in y values of the two points: f(x+h) - f(x).
RUN is the difference in x values of the two points: (x-h) - x. (This is just h.)
Thinking about this function in this way helps you see that the intimidating limit equation is truly just a basic slope function using arbitrary points.