I think the idea is that using the splitting principle everything reduces to the first Chern class of line bundles: Chern classes of a general bundle $E$ are symetric functions of $c_1(L_i)$ where $\bigoplus L_i = E$.
If $Q(z)$ is a power series with constant term 1, you can define $K_n$ by the formula:
$$
\sum K_n(x_1,\ldots,x_n) = \prod Q(z_j)
$$
where $x_i$ is the $i$-th elementary symetric function of the variables $z_j$. Hogomeneity corresponds to the fact that the $z_j$ have degree 1.
I think the statement that every multiplicative sequence comes from such a power series is only true in caracteristic 0. A multiplicative sequence with coefficients in $A$ corresponds to a ring homomorphism from the Lazard ring $\mathbb{L} = \Omega^*(pt)$ to $A$ that is to a formal group law $F(t_1,t_2) \in A[[t_1,t_2]]$. There is a natural action of power series $f(z)$ satistfying $f(0) = 0$ and $f(z) f'(0) = 1$ on the Lazard ring; just change of coordinate of formal group laws $(F^f)(t_1,t_2) := f^{-1}(F(f(t_1),f(t_2))$. Now in caracteristic zero, this action is simply transitive: every law is equivalent to the additive one $(t_1,t_2) \mapsto t_1+t_2$ because we can define the logarithm of a law by formally integrating an invariant differential. This should correspond to the fact that every multiplicative sequence is defined by a power series $Q(z) = z/f(z)$.
