Explanation of delta method?

Definition: For $\displaystyle A\subset\mathbb{R}, X_n$ are A-valued random variables and $\displaystyle a_n (X_n - \theta) \stackrel{D}{\rightarrow} X$ for some sequence $\displaystyle \{a_n\}_{n\in\mathbb{N}}$ satisfying $\displaystyle a_n \rightarrow \infty$ as $\displaystyle n \rightarrow \infty$. Then for any function $\displaystyle h: A \rightarrow\mathbb{R}$ that is differentiable at $\displaystyle \theta , a_n (h(X_n ) - h(\theta)) \stackrel{D}{\rightarrow} h'(\theta)X$

Could someone explain that definition? I had a look on wikipedia (Delta method - Wikipedia, the free encyclopedia) but it doesn't look like the definition above. I don't understand what it's saying or how to use it, can someone please explain, thanks.