Why can insects adapt to pesticides quickly?

Pesticides are a form of selection in the evolution of insects - not a "natural selection" but a selection pressure nonetheless.

If a pesticide is applied to say an crop of whatever and it kills off 99% of the infecting bugs, this is a success for the farmer in the short-term. However, that 1% of the bugs that survive have some trait that makes them immune to that particular pesticide. So, they reproduce and bingo! - you have a new generation of bugs that are immune to the pesticide and they breed like crazy and take over the fields.

Insects have short lifespans compared to other organisms, meaning the reproduce and die relatively quickly. For example, ladybugs (Harmonia axyridis) typically live only three months but females produce an average of 25 eggs per day (source).

This short lifespan and selection for individuals that are resistant to the pesticide means that the farmer and scientists must now find a new pesticide that the new generation of bugs are not immune to and so the cycle goes on and on....in the mean time, a lot of chemicals get added to the environment that end up in the systems of other organisms (including us).

This is known as an evolutionary arms race. In insects, this might occur on a 2-5 year cycle or something like this, while in bacteria that reproduce even faster, this can happen over the course of a few months to up to a year.