I can't seem to understand how (5.3.23) was reached (below). In particular, how is the fact that $\lambda \mapsto \lambda e^{-\lambda/2}$ is decreasing being used?
Here is the passage in question:
For all $k \le n/2$ with $k \ge 5$, we bound $k(n-k) \ge kn/2$, so that $$\mathbb E_\lambda[X_k] \le n(e \lambda e^{-\lambda/2})^k. \tag{5.3.22}$$ As a result, for $\lambda = a \log n$ with $a > 1/2$, and all $k \ge 5$, and using that $\lambda \mapsto \lambda e^{-\lambda/2}$ is decreasing for $\lambda \ge 2$, $$\mathbb E_\lambda[X_k] \le n^{1-k/4}. \tag{5.3.23}$$
If $\lambda = a \log n$, then $e^{-\lambda/2} = n^{-a/2} = \frac{n^{-1/4}}{n^{a/2 - 1/4}},$ and we can write $$e\lambda e^{-\lambda/2} = \frac{a e \log n}{n^{a/2 - 1/4}} \cdot n^{-1/4}.$$ There's that messy factor out front that we want to ignore, but the numerator in it is $O(\log n)$ and the denominator is $n$ raised to a positive power, so when $n$ is large enough the messy factor is less than $1.$
So we know that if $\lambda$ actually is $a \log n$ from some constant $a>\frac12$, then (ignoring the messy factor, as we've earned the right to do) $$\mathbb E_\lambda[X_k] \le n\left(n^{-1/4}\right)^k \le n^{1 - k/4}.$$ As far as I can tell, the point of saying that $\lambda \mapsto \lambda e^{-\lambda/2}$ is decreasing is that Proposition 5.10 wants to be true for all $\lambda \ge a \log n$ for some $a > \frac12$. So if it were true for $\lambda = \frac23 \log n$, but then stopped being true for $\lambda = (\log n)^2$, that would be a problem. It's a problem we don't have here, because making $\lambda$ bigger only makes the term raised to the $k$-th power smaller.