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Do you have questions to Hartshorne's proof or just how to deduce Silverman's result from it?

You can factor the field extension of the function field into a purely separable and purely inseparable extension, so WLOG $\phi$ is separable as a purely inseparable morphism is a universal homeomorphism. As $f$ is finite, it is affine, so it looks locally like $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$. As $\phi$ is separable, the discriminant of $B/A$ is $\neq 0$, which gives us that $f$ is unramified outside a finite set of points (the primes which don't divide the discriminant).

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Do you have questions to Hartshorne's proof or just how to deduce Silverman's result from it?

You can factor the field extension of the function field into a purely separable and purely inseparable extension, so WLOG $\phi$ is separable. As $f$ is finite, it is affine, so it looks locally like $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$. As $\phi$ is separable, the discriminant of $B/A$ is $\neq 0$, which gives us that $f$ is unramified outside a finite set of points (the primes which don't divide the discriminant).