Like Alison said, one can identify $Spec(k^{\mathbb{N}})$ with the set of ultrafilters on $\mathbb{N}$. There is a canonical topology on this set, which makes it into the Stone-Cech compactification of $\mathbb{N}$, $S \mathbb{N}$: one takes as a basis the sets $U_A = ${$ F \in S \mathbb{N} : A \in F $} , where $A \subset \mathbb{N}$.

$S \mathbb{N}$ is universal among compactifications of $ \mathbb{N}$, in the sense that every map from $\mathbb{N}$ to compact $X$ extends uniquely to $S\mathbb{N}$.
It's not hard to see that $S \mathbb{N}$ is homeomorphic to $(Spec(k^{\mathbb{N}}), zariski)$.

I think that what Mumford is pointing at are Stone-Cech compactifications in general rather than $Spec(k^{\mathbb{N}})$ in particular. There's quite a good brief definition and explanation of them in Steen & Seebach's `Counterexamples in Topology'; you could also check out the references on the Wikipedia page.