It is true that there are more functions $\mathbb{R}\to\mathbb{R}$ than there are functions $\mathbb{N}\to\mathbb{R}$. However, the key restriction here is that we are only considering *continuous* functions $\mathbb{R}\to\mathbb{R}$. Very few (relatively speaking) functions $\mathbb{R}\to\mathbb{R}$ are continuous, so few that in fact they can inject into the set of functions $\mathbb{N}\to\mathbb{R}$. A more elementary way to see this than Fourier analysis is to just notice that a continuous function $\mathbb{R}\to\mathbb{R}$ is uniquely determined by its restriction to $\mathbb{Q}$. So restriction gives an injection from the set of continuous functions $\mathbb{R}\to\mathbb{R}$ to the set of functions $\mathbb{Q}\to\mathbb{R}$, which can be identified with the set of functions $\mathbb{N}\to\mathbb{R}$ since $\mathbb{Q}$ is countable.

For some related discussion (and in particular, proofs that all of these sets actually have the same cardinality as $\mathbb{R}$ itself), see Cardinality of set of real continuous functions.