Bézout's Theorem

Let $F(x,y,z)$ and $G(x,y,z)$ be the homogeneous polynomials of degrees $d_1 = \deg(C_1)$ and $d_2 = \deg(C_2)$ defining $C_1$ and $C_2$. Choose projective coordinates so that the point $[0:1:0]$ lies on neither curve and so that no two intersection points share the same $[x:z]$-line through $[0:1:0]$.

Regard $F$ and $G$ as polynomials in the single variable $y$ with coefficients that are homogeneous polynomials in $(x,z)$. Their Sylvester resultant $R(x,z) = \operatorname{Res}_y(F, G)$ is a nonzero homogeneous polynomial in $(x,z)$ (nonzero because $F$ and $G$ share no common factor), and homogeneity calculation on the Sylvester matrix shows that $R(x,z)$ has degree exactly $d_1 d_2$.

Over the algebraically closed field $k$, any homogeneous polynomial in two variables of degree $d_1 d_2$ factors completely into $d_1 d_2$ linear forms, counted with multiplicity. Each linear factor corresponds to a line through $[0:1:0]$ containing a common zero of $F$ and $G$, and the multiplicity of the factor matches the local intersection multiplicity $\dim_k \mathcal{O}_{P}(C_1 \cap C_2)$ at that point, giving the total $\left| C_1 \cap C_2 \right| = \deg(C_1) \cdot \deg(C_2)$.