$L/K$ be an extension of number fields.
Following description is from Gerald J.Janusz "Algebraic Number Fields" p.110.
Let $\tau\colon K\to\mathbb{C}$ be an embedding (=ring homomorphism). Since $\mathbb{C}$ is algebraically closed, we know from Galois theory that there are $g=[L:K]$ embeddings $\tau_i\colon L\to\mathbb{C}$ such that $\tau_i|_K=\tau$. (The $\tau_i$ are distinct and no two can be conjugate (else they could not agree on $K$).)
I don't understand this. What does "Galois theory" indicate?
I think there is a typo in the statement. It should say
Since $\mathbb C$ is algebraically closed, we know from Galois theory that there are $g=[L:K]$ embeddings $\tau_i:{\color{red}L} \to \mathbb C$ such that $\tau_i|K=\tau$.
What it means Whenever when you $K \hookrightarrow L$ an extension of number fields, then any embedding of $K$ into $\mathbb C$ can be extended to exactly $[L:K]$ embeddings of $L$ into $\mathbb C$.
Foe example consider any embedding of $K=\mathbb Q[\sqrt[3]{2}] $ into $\mathbb C$.
Take $L: Q[\sqrt[3]{2},\sqrt{2}]$. Then $[L:K]=2$, and there are exactly two extensions of the above embedding: one which takes $\sqrt{2} \to \sqrt{2}$ and the second one which takes $\sqrt{2} \to -\sqrt{2}$.