Let $X\subset A^n$ be an affine variety, and let $a\in X$ be a point. Show that $\mathscr{C}_{X,a}\cong\mathscr{C}_{A^n,a}/ I(X)\mathscr{C}_{A^n,a}$, where $I(X)\mathscr{C}_{A^n,a}$ denotes the ideal in $\mathscr{C}_{A^n,a}$ generated by all quotients $\frac{f}{1}$ for $f\in I(X)$.
$\mathscr{C}_{X,a}$ is the stalk of $\mathscr{C}_X$ at $a$ and $\mathscr{C}_X$ is the set of all regular functions on $X$.
This problem is very confusing to me, since I don't quite understand the concepts of sheaf and especially stalk.
I should define a map from $\mathscr{C}_{X,a}$ to $\mathscr{C}_{A^n,a}/ I(X)\mathscr{C}_{A^n,a}$:
$$\frac{g+I(X)}{f+I(X)} \mapsto \frac{g}{f} + I(X)\mathscr{C}_{A^n,a}$$
where $f+I(X),g+I(X)\in A(X)$ and $f(a)\neq 0$. I should then prove it is well-defined, injective and surjective. But I cannot continue because I am not sure what right hand side is.
Thanks for any help!