Simple - because your floating island has an underside. Stuff is not just going to be living and growing on the top of it. Plants will be growing in cracks in the rocks underneath, or dangling off the side like a curtain. It won't be quite as diverse as up top, because of the tendency for soil and water to plummet to the ground! Might be a water shortage, since it has the island above acting as an umbrella to keep off the rain. But there WILL be life there.
Any creature which can climb or fly can exploit this underside resources. Birds will be nesting on ledges on the underside (like seabirds on cliffs) or gluing their nests to it (like swallows and swifts), bats will be roosting in caves.
If a climber heads down to gather bird eggs and slips and falls, they are dead. Even if they have a soft landing, they fall off the island and are never seen again. So being able to fly is a lifesaver. There will be a big evolutionary pressure towards flight (and also towards 'not falling off' adaptations, like spiders making safety lines with their silk).
Alternatively... your species has no predators, so can only die of accident, old age, disease or murder. Unless disease and murder are rife they (or their animal ancestors) are going to breed and breed and breed until they overpopulate the island. So there will always be pressure for some of the population to disperse to new islands.
If the other islands conveniently 'dock' with the original now and then, no problem. But if the other islands never touch, they are going to have to jump. Again natural selection will favour creatures who arrive in their new home without broken bones or as a messy splat. Jumping ability, then gliding ability then true flying ability.
Once they can fly, they can colonise islands that never come within jumping or gliding distance - for instance one which is floating far above their current home.