I'm looking for a "conceptual" explanation to the question in the title. The standard proofs that I've seen go as follows: use the Schubert cell decomposition to get a basis for cohomology and show that the special Schubert classes satisfy Pieri's formula. Then use the fact that basic homogeneous symmetric functions $h_n$ are algebraically independent generators of the ring of symmetric functions, to get a surjective homomorphism from the ring of symmetric functions to the cohomology ring. Pieri's rule can be shown with some calculations, but is there any reason a priori to believe that tensor product multiplicities for the general linear group should have anything at all to do with the Grassmannian?
Maybe a more specific question: is it possible to prove that these coefficients are the same without calculating them beforehand?
One motivation for asking is that the cohomology ring of the type B and type C Grassmannians ${\bf OGr}(n,2n+1)$ and ${\bf IGr}(n,2n)$ are described by (modified) Schur P- and Q-functions which seem to have nothing(?) to do with the representation theory of the orthogonal and symplectic groups ${\bf SO}(2n+1)$ and ${\bf Sp}(2n)$. So as far as I can tell the answer isn't just because general linear groups and Grassmannians are "type A" objects.
Best Answer
There are several rings-with-bases to get straight here. I'll explain that, then describe three serious connections (not just
Ehresmann'sLesieur's proof as recounted in the OP).The wrong one is $Rep(GL_d)$, whose basis is indexed by decreasing sequences in ${\mathbb Z}^d$.
That has a subring $Rep(M_d)$, representations of the Lie monoid of all $d\times d$ matrices, whose basis is indexed by decreasing sequences in ${\mathbb N}^d$, or partitions with at most $d$ rows.
That is a quotient of $Rep({\bf Vec})$, the Grothendieck ring of algebraic endofunctors of ${\bf Vec}$, whose basis (coming from Schur functors) is indexed by all partitions. Obviously any such functor will restrict to a rep of $M_d$ (not just $GL_d$); what's amazing is that the irreps either restrict to $0$ (if they have too many rows) or again to irreps!
Harry Tamvakis' proof is to define a natural ring homomorphism $Rep({\bf Vec}) \to H^*(Gr(d,\infty))$, applying a functor to the tautological vector bundle, then doing a Chern-Weil trick to obtain a cohomology class. (It's not just the Euler class of the resulting huge vector bundle.) The Chern-Weil theorem is essentially the statement that Harry's map takes alternating powers to special Schubert classes. So then it must do the right thing, but to know that he essentially repeats the Ehresmann proof.
Kostant studied $H^* (G/P)$ in general, in "Lie algebra cohomology and generalized Schubert cells", by passing to the compact picture $H^* (K/L)$, then to de Rham cohomology, then taking $K$-invariant forms, which means $L$-invariant forms on the tangent space $Lie(K)/Lie(L)$. Then he complexifies that space to $Lie(G)/Lie(L_C)$, and identifies that with $n_+ \oplus n_-$, where $n_+$ is the nilpotent radical of $Lie(P)$. Therefore forms on that space is $Alt^* (n_+) \otimes Alt^* (n_-)$.
Now, there are two things left to do to relate this space to $H^* (G/P)$. One is to take cohomology of this complex (which is hard, but he describes the differential), and the other is to take $L$-invariants as I said. Luckily those commute. Kostant degenerates the differential so as to make sense on each factor separately (at the cost of not quite getting $H^* (G/P)$).
Theorem: (1) Once you take cohomology, $Alt^* (n_+)$ is a multiplicity-free $L$-representation. So when you tensor it with its dual and take $L$-invariants, you get a canonical basis by Schur's lemma. (2) This basis is the degeneration of the Schubert basis.
Theorem: (1) If $P$ is (co?)minuscule, the differential is zero, so you can skip the take-cohomology step. That is, $Alt^* (n_+)$ is already a multiplicity-free $L$-rep. The Schur's lemma basis has structure constants coming from representation theory. (2) In the Grassmannian case, the degeneration doesn't actually affect the answer, so the product of Schubert classes does indeed come from representation theory.
I believe the degenerate product on $H^*(G/P)$ is exactly the one described by [Belkale-Kumar].
It's fun to see what's going on in the Grassmannian case -- $L = U(d) \times U(n-d)$, $n_+ = M_{d,n-d}$, and $Alt^* (n_+)$ contains each partition (or rather, the $U(d)$-irrep corresponding) fitting inside that rectangle tensor its transpose (or rather, the $U(n-d)$-irrep).
I think this is going to be the closest to what you want, for other groups' Grassmannians.