Suppose $X$ is a smooth proper geometrically connected curve defined over a number field $k$. Suppose $f:X\to\mathbb{P}^1_k$ is a degree $d$ map. This means that after assigning coordinates $x$ and $y$ to $\mathbb{P}^1$ and letting $T=x/y$, $[k(T):K(X)]=d$, where $K(X)$ is the function field of $X$. I'd like to know why this means that $X$ has infinitely many points of degree $d$, where we say that $P$ is a point of degree $d$ if $[K(P):K]=d$, where $K(P)$ is the residue field of $P$. For instance, I'm not sure why this might suggest the fibers are of degree $d$. What am I missing?
Edit: We assume $k$ is a number field
Let $f:X\to \mathbb{P}^1_k$ be a finite surjective morphism of degree $d$, where $X$ is a smooth projective geometrically connected curve over $k$.
Hilbert's irreducibility theorem says that the set of $t\in \mathbb{P}^1(k)$ such that $X_t$ is integral is dense.
Now, if the fibre $X_t$ is integral, then $X_t = \mathrm{Spec} L$ with $L/k$ some finite field extension of degree $d$.