Two things drive adaptive radiations: 1) Vacancies, when many ecological niches are empty due to catastrophes or to new creation of land, and 2) Evolutionary innovations, when creatures have new features which give them a major advantage and allow them to drive many competitors out of many ecological niches or move to new environments where there are fewer competitors.
Work Step by Step
See p.536. The basic idea in an adaptive radiation is that descendants of one species or other group take over many ecological niches. For this to happen, either they must walk into an empty niche or there must be some evolutionary reason (selective advantage--review the basics of evolution) for their being able to take over a niche.