Maybe Segal's fascinating extension of this fact to the K-homology adds some intuitive understanding of what happens underneath. Unfortunately I only was able to find a paid version of the text ("K-homology theory and algebraic K-theory"), it is in the book "K-theory and operator algebras" (Springer LNM 575, pp 113-127)
In Segal's setup $X$ is any compact Hausdorff space with a basepoint. He takes its Gelfand dual $C(X)$ (continuous real-valued functions on $X$ vanishing at the basepoint, a (unitless) C-algebra). Recall that Gelfand duality recovers $X$ from $C(X)$ as the spectrum of the latter. That is, every C-homomorphism $C(X)\to\mathbb R$ has form $f\mapsto f(x)$ for some (fixed) $x\in X$.
Segal considers $$ F(X):=\bigcup_{n\geqslant0}\mathrm{Hom}_{\textrm{Algebras}}(C(X),\mathrm{Mat}_{n\times n}\mathbb R), $$ "a kind of non-abelian spectrum of $C(X)$" (union is wrt embedding $n$-matrices into $n+1$-matrices via $A\mapsto\left(\begin{smallmatrix}A&0\\0&0\end{smallmatrix}\right))$.
What matters is that an element of $F(X)$ can be viewed as a "finitely supported family of real f.d. vector spaces $V_x$ indexed by points of $x$"; moreover there is a natural topology on $F(X)$ such that (a) if points $x_1$ and $x_2$ are moved towards each other to coincide in $x$ then $V_x$ becomes identified with the resulting limit of $V_{x_1}\oplus V_{x_2}$; (b) if a point $x$ moves towards the basepoint it just falls out of the picture.
Thus a point of $F(X)$ is like a nonnegative linear combination of points of $X$, except that mutliplicities of points are, instead of natural numbers, finite-dimensional real vector spaces.
It turns out that $\pi_*(F(X))$ is isomorphic to $\widetilde{\mathrm{kO}}_*(X)$ (reduced connective $K$-homology of $X$).
Furthermore, there is a natural map $F(X)\to SP(X)$ sending "$V_1x_1+...+V_kx_k$" to $\dim(V_1)x_1+...+\dim(V_k)x_k$ and the induced map of homotopy groups $\widetilde{\mathrm{kO}}_*(X)\to\tilde H_*(X)$ is the one you've just guessed.