How can adaptive radiation occur on an island chain?
Adaptive radiation occurs on islands the same way it occurs in any area where there is an ecosystem or ecological niche to be exploited.
Adaptive radiation, or when new organisms diversify rapidly due to new environmental pressures, opportunities, or resources, occurs in situations such as after a mass extinction, when a lake has newly formed, after a volcanic explosion drastically changing the landscape, or the formation of new islands.
Islands are particularly good places to look for adaptive radiation because islands are typically isolated and may or may not be entirely new geological entities. If there are ecological niches to be exploited, this creates the opportunity for adaptive radiation. Islands far from the mainland may be very hard to get to, thus limiting the number of species that has made it to the island. This can leave niches unfilled.
For example, the Hawaiian honeycreepers (a subfamily of birds), exhibits a great diversity on the Hawaiian islands. The ancestor of these birds colonized the islands and procreated and diversified rapidly to fill multiple niches: some consume fruit, others seeds, and some feed on nectar. Read more about their evolution here.
Below is an example of speciation of finches on the Galapagos: