Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?

I don't know of a DAG mechanism that implies these statements, any more formally than the standard proofs -- i.e., we reduce to a maximal torus, where the statement follows from the shape of the cohomology of the circle.

There are higher mechanisms though of which this statement is a hint. On one side you have topology (of BG), on the other side (derived) algebraic geometry -- suggesting that rather than something formal this is a shadow of mirror symmetry. Indeed the statement is [essentially equivalent to] the mirror symmetry between the G-equivariant A-model on a point and the B-model into a shift of the Cartan mod Weyl group. That is probably a natural setting for this question.

Or if you prefer, the statement can be seen (at the risk of tremendous overkill) as a feature of geometric Langlands (roughly the case of ${\mathbf P}^1$), specifically of the wonderful derived Geometric Satake theorem of Bezrukavnikov-Finkelberg https://arxiv.org/abs/0707.3799 (though of course the basic statements above are used in the proof!) Namely there's an equivalence of monoidal dg categories

$$D_{LG_+}(LG/LG_+) \simeq D_{coh}({\mathfrak g}^{\vee,\ast}[2]/G^\vee)$$

between equivariant sheaves on the affine Grassmannian for $G$ and equivariant coherent sheaves on the coadjoint representation of the Langlands dual. Your statement is an "affinization" of this -- describing modules for the endomorphisms of the unit (or natural coefficient rings) on both sides -- on the left there's the equivariant category

$$D_{LG_+}(pt)=D_G(pt)=D(BG)$$

and on the right there's the affinization of the coadjoint quotient,

$${\frak g}^{\vee,\ast}[2]//G^\vee \simeq {\frak h}^{\vee,\ast}[2]//W \simeq {\frak h}[2]//W\simeq {\frak g}[2]//G.$$

I believe you can pass from one statement to the other through Koszul duality and formality. More precisely, the cochains of the classifying space and the chains of the group are Koszul dual. So once you know that the cochains are quasisomorphic to the $G$ invariants of a polynomial ring, you get that the chains are quasisomorphic to the $G$ invariants of an exterior algebra.