More generally, if $F$ is a totally real number field with ring of integers $\mathfrak{o}$, then Lichtenbaum (and Quillen) conjectured that $|\zeta_F(1-i)|=\frac{\# K_{2i-2}(\mathfrak{o})}{\# K_{2i-1}(\mathfrak{o})}$, times a regulator and some power of 2 (which I believe is not understood in general, although some progress was made on this in Ion Rada's PhD thesis). Hence, odd $K$ groups are related to the denominators of the Bernoulli numbers, and the even ones are related to the numerators. Also, not much cancellation occurs; I think the two $K$-groups can only share factors of 2.
The Voevodsky-Rost theorem might prove the Lichtenbaum conjecture, but I haven't seen anyone come out and say definitely that this is the case.
I don't have much intuition for this, except that the $K$-groups seem to be objects that like to map into étale cohomology groups. In this paper (link to MathSciNet), Soulé constructs Chern class maps from certain $K$-groups to étale cohomology groups. Furthermore, these maps frequently have small (or trivial) kernels and cokernels. I suppose the idea, then, is that $K$-theory is supposed to be a slightly better behaved version of étale cohomology, at least for the purpose of understanding zeta functions.
The rank of $K$-groups of rings of integers was computed by Quillen in the early 70's: it's rank 1 in dimension 0, rank $r_1+r_2-1$ in dimension 1 (Dirichlet's unit theorem), rank 0 in even dimensions $>0$, rank $r_1+r_2$ in dimensions $1\pmod 4$ except 1, and rank $r_2$ in dimensions $3\pmod 4$.