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.**