Stating it first for clarity for N2, for any pair (m,n) of natural numbers we can introduce Rm,n, the 'rectangle' of numbers (r, s) with r at least m and s at least n. This is semi-infinite in the north and east directions, in the usual plane representation. The lemma then states that any union of the Rm,n is a finite union.
The generalization to Nk is the natural one, with k-tuples in place of pairs.
The statement says something about Nk as the topological space with the product topology arising from N, where the latter has the (semi-continuity) topology in which the open sets are all sets Rm defined as all n with n at least m. The 'rectangles' are by definition a base for the topology; it says finite unions give all open sets.
As for the proof of the lemma, it can be derived directly, but a slick way is to show that it is a special case of Hilbert's basis theorem - in fact is essentially the case of ideals generated by monomials.