From Silverman and Tate Rational Points on Elliptic Curves, ch. 6
$K$ is a number field, we are looking at the set of field homomorphisms $$\sigma: K \hookrightarrow \mathbb{C}.$$
We recall that a homomorphism of fields is always one-to-one because a field has no non-trivial ideals. Also, since by definition $\sigma(1) = 1$, we see that $\sigma(a)=a $ for all $a \in \mathbb{Q}$. It is a theorem that the number of homomorphisms $K \hookrightarrow \mathbb{C}$ is exactly equal to the degree $[K : \mathbb{Q} ]$.
Why are these last two sentences not contradictory? Doesn’t
$\sigma(1) = 1$, so $\sigma(a)=a $ for all $a \in \mathbb{Q}$
imply that there is only one homomorphism?
No, the statements you mentioned are not contradictory. I will give an example where the number field $K$ is $\mathbb{Q}(e^{2\pi i/p})$ (using what I know from Marcus' Number Fields) to illustrate and elaborate further on what @CaptainLama has said. Should anyone find an error in my explanation, feel free to correct me, here or in the comments.
We recall that a number field $K$ is a finite degree field extension of $\mathbb{Q}$. Let $p$ be a prime. Then $\mathbb{Q}(e^{2\pi i/p})$ is a number field of degree $\varphi(p)=p-1$ over $\mathbb{Q}$, i.e. $[\mathbb{Q}(e^{2\pi i/p}):\mathbb{Q}]=\varphi(p)=p-1$, where $\varphi$ is the Euler $\varphi$-function.
The homomorphisms you are referring to are embeddings (injective homomorphisms) from the number field $K (=\mathbb{Q}(e^{2\pi i/p}))$ into $\mathbb{C}$ which fixes $\mathbb{Q}$ pointwise. The phrase ``$\sigma$ fixes $\mathbb{Q}$ pointwise'' is used in the sense that an embedding \begin{align*} \sigma:K=\mathbb{Q}(e^{2\pi i/p})\hookrightarrow\mathbb{C} \end{align*} is such that $\sigma(a)=a$ for all $a\in\mathbb{Q}$. But we know that there is a strict inclusion (proper subset) of $\mathbb{Q}\subsetneq\mathbb{Q}(e^{2\pi i/p})$. So in that sense, it is incorrect to say that we know the values of $\sigma$ on $K$ simply by its values on $\mathbb{Q}\subsetneq K$, since there are values of $\sigma$ in $K\setminus\mathbb{Q}\neq\emptyset$ that we also need to determine. That is to say, we also need to determine the values of $\sigma$ on $\mathbb{Q}(e^{2\pi i/p})\setminus\mathbb{Q}\neq\emptyset$ (for instance, the value of $\sigma$ on $e^{2\pi i/p}$). Moreover, from Field Theory, we know that a field homomorphism $\tau:\mathbb{Q}(\alpha)\rightarrow\mathbb{Q}(\alpha)$ is determined by its values on $\mathbb{Q}$ and $\alpha$, where $\alpha$ is an algebraic integer.
If we denote $\zeta_{p}=e^{2\pi i/p}$ by the $p$-th root of unity, we see that we get $p-1$ distinct embeddings from $K$ into $\mathbb{C}$, namely, $\sigma_{j}(\zeta_{p})=\zeta_{p}^{j}$ for $j=1,2,\dots,p-1$. One can check that each $\sigma_{j}$ is indeed an injective homomorphism from $K=\mathbb{Q}(\zeta_{p})$ to $\mathbb{C}$.