Just for completeness, here's another argument without spectral sequences via rational homotopy theory.

Recall a theorem of Hopf, which states that the rational cohomology of a path-connected H-space of finite rational cohomology type(finite dimensional rational cohomology in each degree) is a free graded commutative graded algebra (cga) $(\wedge V,0)$, since it's a Hopf algebra. This applies to a compact lie group $G$ and implies immediately, that the rational cohomology serves as a minimal model of $G$ and since the cohomology of a compact Lie group is finite dimensional, $V$ has to be in odd degrees.

The long exact sequence of the universal fibration $G\rightarrow EG\rightarrow BG$ shows, that $\pi_i(BG)\cong\pi_{i-1}(G)$ since $EG$ is contractible. Recall that the minimal model of a space $X$ is a cdga whose underlying cga is the free cga generated by $\pi_*(X)\otimes\mathbb{Q}$ with a differential $d$, i.e. has the shape $(\wedge(\pi_*(X)\otimes\mathbb{Q}),d)$, so the minimal model of $BG$ has the shape $(\wedge(\pi_{*+1}(G)\otimes\mathbb{Q}),d)$. Since the minimal model of $G$ is concentrated in odd degrees, the one of $BG$ is concentrated in even degrees, so the differentailo must vanish for degree reasons and the minimal model of $B$ is just $(\wedge(\pi_{*+1}(G)\otimes\mathbb{Q}),0)$. Since the cohomology of the minimal model of a space is the cohomology of the space, we get the claim.

Just for completeness, here's another argument without spectral sequences via rational homotopy theory.

Recall a theorem of Hopf, which states that the rational cohomology of a path-connected H-space of finite rational cohomology type  (finite dimensional rational cohomology in each degree) is a free graded commutative graded algebra (cga) $(\wedge V,0)$ (The reason is, that the cohomology has a Hopf algebra structure.) This applies to a compact lie group $G$ and implies immediately, that the rational cohomology serves as a minimal model of $G$ and since the cohomology of a compact Lie group is finite dimensional, $V$ has to be concentrated in odd degrees.

The long exact sequence of the universal fibration $G\rightarrow EG\rightarrow BG$ shows, that $\pi_i(BG)\cong\pi_{i-1}(G)$ since $EG$ is contractible. Recall that the minimal model of a space $X$ is a cdga whose underlying cga is the free cga generated by $\pi_*(X)\otimes\mathbb{Q}$ with a differential $d$, i.e. has the shape $(\wedge(\pi_*(X)\otimes\mathbb{Q}),d)$, so the minimal model of $BG$ has the shape $(\wedge(\pi_{*+1}(G)\otimes\mathbb{Q}),d)$. Since the minimal model of $G$ is concentrated in odd degrees, the one of $BG$ is concentrated in even degrees, so the differential must vanish for degree reasons and the minimal model of $BG$ is just $(\wedge(\pi_{*+1}(G)\otimes\mathbb{Q}),0)$. Since the cohomology of the minimal model of a space is the cohomology of the space, we get the claim.