He writes
Let $W\subset U$ be a rectangle of the form $D\times [a_n,b_n]$, where $D $ is rectangle in $\mathbb R^n$. By Fubini's thoerem $$\int _{h(W)}1=\int _{[a_n,b_n]} {\Bigg(} \int _{h(D\times \{x^n\})}1 dx^1 \cdots dx^{n-1}{\Bigg ) }dx^n$$
However shouldn't this integral be $$\int _{h(W)}1=\int _{h([a_n,b_n])} {\Bigg(} \int _{h(D)}1 dx^1 \cdots dx^{n-1}{\Bigg ) }dx^n $$ How do we get from this one to the above one?
No, I think it is correct. Think of this in the sense of summing things up: For a fixed $t \in [a,b]$ he integrates over $h(D \times \{t\})$, which is in his context a single rectangle of the "familiy of parametrized" rectangles. Now to account for all possible variations, he still needs to integrate over all possible values for $t$, hence the outer integral.
In other words, $h(D \times \{t\})$ is a cut through $h(D \times [a,b])$ reducing the dimensionality of $h(D \times [a,b])$ by one (assuming that $h$ has "nice" behaviour). After getting the volume of this, you still need to sum things up over all $t \in [a,b]$.