Let $C$ and $D$ be complex plane affine algebraic curves defined by polynomials $f(x,y)$ and $g(x,y)$. One of the definitions for intersection multiplicity of $F$ and $G$ at some point $P$ is $$i(C,D,P) = \dim \mathcal{O}_{\mathbb{C}^2,P}/(f,g)$$ (where $ \mathcal{O}_{\mathbb{C}^2,P}$ is the local ring of $\mathbb{C}^2$ at the point $P$). For our case, we can assume, that $P=(0,0)$.
If $D$ is parametrized by $\varphi(t) : \mathbb{C}^2 \rightarrow \mathbb{C}$, and $\varphi(0)=(0,0)$, where $\varphi$ is injective on some neighbourhood of $(0,0)$, the intersection multiplicity of $C$ and $D$ at $(0,0)$ is also equal to the order of the polynomial $f(\varphi(t))$.
Question: What is the connection between these two definitions? Why are they equal?
(Both intuitive explanation or the correct proof would be helpful to me)