Most of this is not special to the case of $AG_{\infty,2}^+$. For any spectrum $X$, we have a Hurewicz map $h:\pi_{\ast}(X)\to H_{\ast}(X)$, which induces a map $h':\mathbb{Q}\otimes\pi_{\ast}(X)\to\mathbb{Q}\otimes H_{\ast}(X)$. Standard calculations show that $h'$ this is an isomorphism when $X=S^n$ for some $n$, and it follows by induction up the skeleta that $h'$ is an isomorphism for all $X$. Next, the homotopy groups $\pi_{\ast}(X)$ are (essentially by definition) the same as the homotopy groups of the space $\Omega^\infty(X)$, so we have an unstable Hurewicz map $h'':\pi_{\ast}(X)=\pi_{\ast}(\Omega^\infty(X))\to H_{\ast}(\Omega^\infty(X))$. By combinining these we get a map $\mathbb{Q}\otimes H_\ast(X)\to \mathbb{Q}\otimes H_\ast(\Omega^\infty(X))$. Next, every infinite loop space is a homotopy-commutative H-space, and this makes $H_\ast(\Omega^\infty(X))$ into a graded-commutative (and graded-cocommutative) Hopf algebra. Now let $A_*$ be the free graded-commutative ring generated by $\mathbb{Q}\otimes H_\ast(X)$, with the Hopf algebra structure for which the generators are primitive. More explicitly, $A_\ast$ is a tensor product of polynomial algebras (one for each even-dimensional generator in $\mathbb{Q}\otimes H_\ast(X)$) and exterior algebras (one for each odd-dimensional generator). It is now quite formal to construct a canonical map $A_\ast\to\mathbb{Q}\otimes H_\ast(\Omega^\infty(X))$ of bicommutative Hopf algebras. One can then show that this map is always an isomorphism. Indeed, it is possible to reduce to the case $X=S^n$ again, and the groups $\mathbb{Q}\otimes H_\ast(\Omega^k S^{n+k})$ can be calculated by repeated use of Serre spectral sequences, and one can then let $k$ tend to infinity. As $\mathbb{Q}\otimes H_\ast(\Omega^\infty(X))$ is free on primitive generators, it is a standard fact that the dual Hopf algebra $\mathbb{Q}\otimes H^\ast(\Omega^\infty(X))$ is also free on primitive generators in the same degrees.

For the Madsen-Weiss case, you now just need to know the groups
$\mathbb{Q}\otimes H_\ast(AG_{\infty,2}^+)$. This is not hard, because $AG^+_{\infty,2}$ is the Thom space of a virtual vector bundle over $BSO(2)=\mathbb{C}P^\infty$, so we can use the Thom isomorphism theorem.