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In fact "$\mathbb{Q}$-acyclic" means "having the same rational homology groups as a point". In particular, the classifying space of any finite group is $\mathbb{Q}$-acyclic, as can be seen by a simple argument using the transfer.

Once you know that the fibers of $f\colon\thinspace EG\times_G X\to G\backslash X$ are $\mathbb{Q}$-acyclic, you can conclude that its Leray spectral sequence collapses at the $E_2$-term, giving the desired isomorphism. (Any map $f\colon\thinspace X\to Y$ has a spectral sequence whose $E_2$-page is cohomology of $Y$ with coefficients in the sheaf given by $U\mapsto H^\ast(f^{-1}(U))$, of which the Leray--Serre spectral sequence of a fibration is a special case. See section 14 of Bott and Tu's "Differential forms in algebraic topology" for a nice introduction).

In fact "$\mathbb{Q}$-acyclic" means "having the same rational homology groups as a point". In particular, the classifying space of any finite group is $\mathbb{Q}$-acyclic, as can be seen by a simple argument using the transfer.

Once you know that the fibers of $f\colon\thinspace EG\times_G X\to G\backslash X$ are $\mathbb{Q}$-acyclic, you can conclude that its Leray spectral sequence collapses at the $E_2$-term, giving the desired isomorphism. (Any map $f\colon\thinspace X\to Y$ has a spectral sequence whose $E_2$-page is cohomology of $Y$ with coefficients in the sheaf given by $U\mapsto H^\ast(f^{-1}(U))$, of which the Leray--Serre spectral sequence of a fibration is a special case. See section 14 of Bott and Tu's "Differential forms in algebraic topology" for a nice introduction).

Edit: As others have noted, the spectral sequence does not collapse without additional assumptions. In my head when writing this answer I had $G$ a compact Lie group acting smoothly on a smooth manifold $X$. Then the spectral sequence does collapse (as noted by Johannes', since tubular $G$-neighbourhoods of orbits furnish $X/G$ with a good cover) but its probably easier to argue as in Ralph's answer, using a compact approximation to $E_G$ and the Vietoris-Begle theorem.

In fact "$\mathbb{Q}$-acyclic" means "having the same rational homology groups as a point". In particular, the classifying space of any finite group is $\mathbb{Q}$-acyclic, as can be seen by a simple argument using the transfer.

Once you know that the fibers of your map $f\colon\thinspace EG\times_G X\to G\backslash X$ are $\mathbb{Q}$-acyclic, you can apply the Vietoris-Begle mapping theorem. More generally, the map Leray spectral sequence of the map $f$ collapses at the $E_2$-term. (Any map $f\colon\thinspace X\to Y$ has a spectral sequence whose $E_2$-page is cohomology of $Y$ with coefficients in the sheaf given by $U\mapsto H^\ast(f^{-1}(U))$, of which the Leray--Serre spectral sequence of a fibration is a special case. See section 14 of Bott and Tu's "Differential forms in algebraic topology" for a nice introduction).

In fact "$\mathbb{Q}$-acyclic" means "having the same rational homology groups as a point". In particular, the classifying space of any finite group is $\mathbb{Q}$-acyclic, as can be seen by a simple argument using the transfer.

Once you know that the fibers of $f\colon\thinspace EG\times_G X\to G\backslash X$ are $\mathbb{Q}$-acyclic, you can conclude that its Leray spectral sequence collapses at the $E_2$-term, giving the desired isomorphism. (Any map $f\colon\thinspace X\to Y$ has a spectral sequence whose $E_2$-page is cohomology of $Y$ with coefficients in the sheaf given by $U\mapsto H^\ast(f^{-1}(U))$, of which the Leray--Serre spectral sequence of a fibration is a special case. See section 14 of Bott and Tu's "Differential forms in algebraic topology" for a nice introduction).

In fact "$\mathbb{Q}$-acyclic" means "having the same rational homology groups as a point". In particular, the classifying space of any finite group is $\mathbb{Q}$-acyclic, as can be seen by a simple argument using the transfer.

Once you know that the fibers of your map $f\colon\thinspace EG\times_G X\to G\backslash X$ are $\mathbb{Q}$-acyclic, you can apply the Vietoris-Begle mapping theorem. More generally, the map Leray spectral sequence of the map $f$ collapses at the $E_2$-term. (Any map $f\colon\thinspace X\to Y$ has a spectral sequence whose $E_2$-page is cohomology of $Y$ with coefficients in the sheaf given by $U\mapsto H^\ast(f^{-1}(U))$, of which the Leray--Serre spectral sequence of a fibration is a special case. See section 14 of Bott and Tu's "Differential forms in algebraic topology" for a nice introduction).

In fact "$\mathbb{Q}$-acyclic" means "having the same rational homology groups as a point". In particular, the classifying space of any finite group is $\mathbb{Q}$-acyclic, as can be seen by a simple argument using the transfer.

Once you know that the fibers of your map $f\colon\thinspace EG\times_G X\to G\backslash X$ are $\mathbb{Q}$-acyclic, you can apply the Vietoris-Begle mapping theorem. More generally, the map Leray spectral sequence of the map $f$ collapses at the $E_2$-term. (Any map $f\colon\thinspace X\to Y$ has a spectral sequence whose $E_2$-page is cohomology of $Y$ with coefficients in the sheaf given by $U\mapsto H^\ast(f^{-1}(U))$, of which the Leray--Serre spectral sequence of a fibration is a special case. See section 14 of Bott and Tu's "Differential forms in algebraic topology" for a nice introduction).