calculus - Understanding the connection between derivative and increasing of function at points

I'm currently studying about derivatives, and i noticed this theorem:
"Let $f$ be a function defined in a neighborhood of $x_0$. Suppose that $f\:'\left(x_0\right)>0$. So $f$ is strictly increasing at $x_0$".

First, what does it even mean "$f$ strictly increasing at $x_0$"?
Second, can someone provide me a proof of that? I thought about the Mean Value Theorem but i don't know how to start, so maybe i need to use another approach.

Thanks!