Maybe because their babies evolved to cope with comparatively low oxygen environments, such as warm water that isn't moving a whole lot. Bichirs aren't exactly fast moving arctic fish and increasing surface area under the operculum/gill cover is fairly tough if it's packed full so external gills can help. This is fairly consistent with air gulping seen in adults, which may also live in less than turbulent warm water. Source: Just a shoot at the hip take by someone who works in paleontology and has a relatively passable knowledge on evolution and how comparative anatomy can relate to it.
For the record: cold water has a higher capacity to house dissolved oxygen and the reason you want moving water is for increased surface area for oxygen to dissolve in.
Also note that external gills in aquatic amphibians are probably more the result of the skull becoming less kinetic, even compared to bichirs which are comparatively very immobile to telostei and other more advanced fish. That's a whole other thing though.
Also: A search on youtube for adult bichirs with external gills yields nothing.