Because it isn't explained, Dragonfly Fishing is the name for downed-pilot-recovery, which was the role destroyers escorting carriers took. Japanese carriers (especially including Akagi) were pretty terrible about pilots overshooting on landings and having to get rescued.
Hurry up.Howaaargggh!?
#013: Duties of the Special-Type DD
SLIIIDEYou're quite suited to be an escort ship, Fubuki-san.Oh my, dear me, no wonder you're the lead ship of us Special-Type destroyers.I've made a recommendation for you to the Carrier Battle Group.
Kidou Butai.Dragonfly fishing, or retrieving downed planes (and their pilots) that made emergency landings in the water. One of the actual duties of the real-life Fubuki, like most IJN DDs accompanying carrier forces.I'll do my best!Eeeh, really!? I'm going be in the same unit with Akagi this soon!?Hey, Special-Type DD!
Let's go on a night fishing trip!Here.
Your specialized equipment.