Corollary 15.5. Let $S$ be a bounded set in $\mathbb{R}^n$; let $f:S\to\mathbb{R}$ be a bounded continuous function. If $f$ is integrable over $S$ in the ordinary sense, then $$(\text{ordinary})\int_S f=(\text{extended})\int_{\operatorname{Int}S} f.$$
This corollary tells us that any theorem we prove about extended integrals has implications for ordinary integrals. The change of variables theorem, which we prove in the next chapter, is an important example.
I don't understand what the author wants to say in the above two sentences perfectly.
Does the author want to say like the following?:
Let $C$ be a bounded set in $\mathbb{R}^n$.
Let $f:C\to\mathbb{R}$ be a bounded continuous function.
Suppose $f$ is integrable over $C$.
Let $B:=\operatorname{Int}C$.
Let $A$ be an open set in $\mathbb{R}^n$.
Let $g:A\to B$ be a diffeomorphism of open sets in $\mathbb{R}^n$.
Then, by Corollary 15.5 above and by Theorem 17.2 below, $$(\text{ordinary})\int_C f=(\text{extended})\int_B f=(\text{extended})\int_A (f\circ g)|\det Dg|.$$
If it is easy to calculate $(\text{extended})\int_A (f\circ g)|\det Dg|$, then we are glad.
We may apply Theorem 17.2 not only to an open set $B$ but also to $C$ which is not open.
Theorem 17.2 (Change of variables theorem). Let $g:A\to B$ be a diffeomorphism of open sets in $\mathbb{R}^n$. Let $f:B\to\mathbb{R}$ be a continuous function. Then $f$ is integrable over $B$ if and only if the function $(f\circ g)|\det Dg|$ is integrable over $A$; in this case, $$\int_B f=\int_A (f\circ g)|\det Dg|.$$