My two cents.
"feeding", like "hatching", which occurs earlier in the same sentence, is a participle, that is, an adjective derived from a verb by adding "-ing".
There is no uniform paraphrase that can be used for all participles, but here is a possible paraphrase of your example sentence.
All eels spawn in the sea, the eggs hatching into transparent, ribbon-like larvae that drift about, feeding until they metamorphose into small eels.
All eels lay their eggs in the sea, and the eggs hatch into transparent, ribbon-like larvae that drift about, and while the larvae drift about they feed until they change into small eels.
So "feeding" describes "larvae". What kind of larvae are they? They are "feeding larvae". This is another way of saying "larvae that feed".
So the use of participles in this way can be thought of as a stylistic variant of relative clauses.