What they eat varies with species. I think carpet are fish eater. There is a paper some where that notes that there are a lot of scallop shells under one of the anemone species. I highly doubt that Crispa, Malu, Magnifica, Aurora can catch a fish of any size.
In my tank I see anemones, mainly Haddoni and Gigantea, eat a lot of worm that swim around and spawn at night. I have substrate worms in my sand bed that go up at night to spawn. They swim around pretty much with random movement.. If they touch one of my carpets, they are history. My carpets make frequent meals of my snails. The can get rather sick when they eat a large Bristtle worm. I am sure any fish eggs and larvae though my carpet get eaten too. Arthropods with the hard shell seem to be resistant to carpet sting. Shrimps can craw across my carpets without much problem.