More precisely, the intuition can be stated as: if the derivative is positive, there is some point to the right of x_0 where f is greater, and some point to the left of x_0 where f is less, and thus f attains neither a maximum nor a minimum at x_0.
The theorem (and its proof below) is more general than the intuition in that it doesn't require the function to be differentiable over a neighbourhood around \displaystyle x_0.