First, here is what I know:
Some infectious illnesses, such as HIV or Hepatitis B, can be treated with the help of some drugs. The problem is that the virus sometimes undergoes mutations which grant resistance to the drug.
First question : Is it so far right?
Secondly, what is not totally clear is why taking drug does speed up the development of the drug-resistant version of the virus. In my point of view, the drug resistant virus will develop anyway with or without taking drugs. I don't see what taking drugs changes. Is it like an intra-competition between the different mutated versions of the virus and in this case, taking drug will eliminate all the mutated versions that are not drug-resistant and, this way, favour the development of the drug-resistant versions?