Let $Φ:C→D$ be a non constant map of smooth curves. Silverman says in AEC
for all but finitely many $y∈D$, $$\#\phi^{-1}(y) = \deg_s(\phi)$$
How can I prove this? Silverman also say to prove this, see Hartshorne Ⅱ6.8. But I can find no connection between them.
Some remarks in mathstacks I see use the ideal correspondence and Hilbert's Nullstellensatz, but curves $C$ and $D$ in AEC might not be defined over algebraically closed field, so I need to show the proposition over arbitrary fields.
I assume you mean Proposition II.2.6(b) in AEC. The curves must be defined over an algebraically closed field, or else the statement is not true. An easy counterexample in the non-algebraically closed case is $K = \mathbb{R}$, with curves $C = D = \mathbb{A}^1(\mathbb{R})$, and $f(x) = x^2$. Then $\# \phi^{-1}(y) = 0$ for infinitely many $y$ (namely, all $y < 0$), but $0 \neq \deg_s\phi = 2$.
In order to understand how II.2.6(b) incorporates a hypothesis of an algebraically closed field, it is necessary to refer to the Definition of smooth curve (on page 17), which refers to the Definition of projective variety (on page 9), which refers to the Definition of projective algebraic set (on page 7), which refers to the definition of $V_I$ (on page 7), which refers to the Definition of $\mathbb{P}^n$ (on page 6), which refers to the Definition of $\mathbb{A}^n$ (on page 1). We finally see that $\mathbb{A}^n$ by definition means $\mathbb{A}^n(\bar{K})$, so we are indeed working over an algebraic closure.
In order to avoid a notation conflict with $L$ and $K$ below, I will use $F$ to denote the (algebraically closed) scalar field over which $C$ and $D$ and $\phi$ are defined.
To prove the result, use the fact that over an algebraic closure, points on curves are in 1-1 correspondence with nonzero prime ideals in the coordinate ring. The splitting of prime ideals in finite extensions of global fields then proves the desired result. Specifically, take $K = \phi^* F(D)$ to be the function field of $D$, and $L = F(C)$ to be the function field of $C$, and consider the formula $$ [L:K] = \sum_{j=1}^g e_i f_i $$ from the aforementioned page, where $e_i$ and $f_i$ denote ramification index and inertial degree. The residue fields at every point must all be equal to $F$, since $F$ is algebraically closed, meaning that no further nontrivial finite extensions of this residue field are possible. Hence $f_i = 1$ for all $i$, and the formula then reduces to the simpler expression $$ [L:K] = \sum_{j=1}^g e_i. $$ (Note that this expression is identical to the statement of AEC II.2.6(a).) Since $[L:K] = \deg \phi$ by definition of $\deg \phi$, this equation is exactly the statement that $\deg \phi = \# \phi^{-1}(y)$ for all $y \in D$, where the points in $\phi^{-1}(y)$ are counted with multiplicity (the respective values of $e_i$ denoting the multiplicity of each element of $\phi^{-1}(y)$). In order to obtain II.2.6(b) in the separable case, simply observe that $\deg_s\phi = \deg \phi$, and that there are only finitely many points in $C$ where $e_i \neq 1$.
In the inseparable case, you will have $e_i = d$ where $d$ denotes the inseparable degree of $[L:K]$. Dividing both sides of the equation by $d$ then yields the result.
I also highly recommend that you read Remark II.2.8 in AEC since this remark adds some further information to this discussion.