The following is my opinion, please correct it if I'm wrong or not good explanation:
if we get $a, b$, then there must be a constant changing-rate $m$ of $(a, f(a))$ and $(b, f(b))$, if the function go this straight line, then everything point has derivative equal to m, if f'(x) greater or lower than m, then it must be somewhere to lower or greater than m in order to reach the f(b), between them, it meets the m. If we get this 'feeling', then theorem is just natural and obvious : there must exist at least one x for, f'(x)=\frac{f(b)-f(a)}{b-a}=m
I realize if I think like this way to get the mathematical feeling behind definitions or theorems, then most of them are just natural, for example, fundamental theorem of calculus, if a function $f(x)$, we consider function value as changing-rate, $\lim\limits_{n\rightarrow\infty}\left[\sum_{i=1}^{n}f(x_i)(x_{i}-x_{i-1})\right]$ is just how much the original function-value changes, i.e. $\Delta F(x)=F(b)-F(a)$
But for some other things which always called rules, it seems cannot be understood directly, like the Chain rule, I could prove it by basic definition of derivative, but just cannot 'feel' it as the same way with Mean-Value theorem.