Daniel Harlow vs. Don Page: average entropy of a subsystem

Consider a bipartite system $mathcal{H}_{mathcal{A B}}=mathcal{H}_{A} otimes mathcal{H}_{B}$. Let’s take $|A|leq|B|$, where by $|cdot|$ I mean the dimenstionality of the indicated system.

Also, consider a random pure state defined as $|psi(U)rangle equiv Uleft|psi_{0}rightrangle$, where $U$ is the random unitary matrix chosen from group-invariant Haar measure.

Page’s Theorem says that a randomly chosen pure state in $mathcal{H}_{mathcal{A B}}$ is likely to be very close to maximally entangled as long as $frac{|A|}{|B|} ll 1$.

The original proof given by Page is quite laborious, but Daniel Harlow in Jerusalem Lectures on Black Holes and Quantum Information presents more compact form of the proof. It goes as follow

First defining $Delta rho_{A} equiv rho_{A}-frac{I_{A}}{|A|}$. The entanglement entropy
$$
begin{aligned}
int d U S_{A} &=-int d U operatorname{Tr} rho_{A} log rho_{A}
&=operatorname{Tr}left[left(frac{I_{A}}{|A|}+Delta rho_{A}right)left(log |A|-|A| Delta rho_{A}+frac{1}{2}|A|^{2} Delta rho_{A}^{2}+ldotsright)right]
&=log |A|-frac{|A|}{2} int d U operatorname{Tr} Delta rho_{A}^{2}+ldots
&=log |A|-frac{1}{2} frac{|A|}{|B|}+ldots
end{aligned}
$$
where … indicate terms that are smaller in the limit $|A|,|B| gg 1$

My questions are: Is there any book where this calcaulation is presented ‘step-by-step’? I don’t know what is the meaning of integrating by $U$ – is it in some sense averaging over all possibilities? What happens to the integral in the second line? What is the Haar measure?