$BO$ is the underlying space of a spectrum called the real K-theory spectrum. This spectrum represents a cohomology theory, namely real K-theory, and this means that $BO$ has much more structure than an H-space: it is in fact an infinite loop space, which is loosely a homotopy-theoretic version of an abelian group (as opposed to merely a monoid).

If you believe that homotopy classes of maps $X \to BO$ classify stable real vector bundles on $X$, then the H-space structure on $BO$ comes from direct sum of vector bundles.

$BO$ is the connected component of the zeroth space of a spectrum called the real K-theory spectrum. This spectrum represents a cohomology theory, namely real K-theory, and this means that $BO$ has much more structure than an H-space: it is in fact an infinite loop space, which is loosely a homotopy-theoretic version of an abelian group (as opposed to merely a monoid).

If you believe that homotopy classes of maps $X \to BO$ classify stable real vector bundles (ignoring their dimension) on $X$, then the H-space structure on $BO$ comes from direct sum of vector bundles.