Yes, $\mathbb{N}$ means set of natural numbers.

In Rosen, It is defined for Constant Functions.

when We write, $f : \mathbb{N}^{^{k}} \rightarrow \mathbb{N}$

It means domain is taken as tuple of natural numbers whose size is $k$ and $f$ maps this tuple of natural numbers to a natural number.

For eg. $f : \mathbb{N}^{^{2}} \rightarrow \mathbb{N}$ (or) $f : \mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$ then domain $(x,y) \in \mathbb{N} \times \mathbb{N} $ and co-domain $\in \mathbb{N}$

We can take any function of 2 variables as an example for this like $f(x,y) = x^{2} + y^{2}$

Now,

Here, $C_{a}^{k}$ is just a representation for constant function. This function takes a tuple of natural numbers $(x_1,x_2,......x_k)$ as domain and map to a number 'a'. $X$ represents a tuple $(x_1,x_2,......x_k) \in \mathbb{N} \times \mathbb{N}\times.....k\; times $ (or) $X \in \mathbb{N}^{^{k}}$