A proof by contradiction of a statement $B$ is when we start with the assumption of $\lnot B,$ derive something false from that assumption, and then conclude that the assumption must have been wrong (i.e. $B$ is true).
But often times are theorems are if/then statements, in other words of the form $A\to B.$ A proof of $A\to B$ by contrapositive is a proof of $\lnot B\to \lnot A,$ which we typically prove by assuming $\lnot B$ and deriving $\lnot A.$ This is valid since the contrapositive $\lnot B\to \lnot A$ is logically equivalent to $A\to B.$
People often confuse these two and say a proof is 'by contradiction' even if it actually takes the second form. There is a good reason for this. When we're proving $A\to B,$ the way we generally think about it is that we assume $A$ and then prove $B$ under that assumption. One route we might take is to prove $B$ by contradiction by assuming $\lnot B$ and then proving $\lnot A,$ which is a false statement under our assumption of $A.$
Whereas in a proof by contradiction we derive any false statement whatsoever, in the pattern above, we derive a very specific statement $\lnot A$ which is only false under the assumption of $A$ that we made. This is what the author meant by "where the contradiction is with the theorem's hypothesis."
Notice the part where all the work was done was in assuming $\lnot B$ and then deriving $\lnot A,$ i.e. directly proving the contrapositive $\lnot B\to \lnot A$. However, we mentally framed it as a proof by contradiction of $B$, under the assumption $A$, which amounts to a proof of $A\to B$. Note this is not at all the same thing as a true proof by contradiction of the statement $A\to B,$ which would consist of assuming $\lnot(A\to B)$ and deriving a false statement.