Bdelloids reproduce by parthenogenesis: females produce eggs that develop without fertilisation. Long-term asexuality is expected to be an evolutionary dead end, because harmful mutations accumulate without recombination to clear them, so a diverse and ancient asexual clade is a genuine puzzle.
Genome work has offered part of an answer. Bdelloid genomes contain an unusually high proportion of genes apparently acquired from bacteria, fungi and plants, and desiccation repeatedly fragments their DNA and forces repair. Both features may substitute for recombination by shuffling genetic material from outside and within.
The honest position is that the puzzle is not closed. Some researchers argue for rare or cryptic forms of genetic exchange that would make the group not truly asexual, and evidence for horizontal transfer has been challenged on contamination grounds in individual studies. What is not in dispute is that no male bdelloid has ever been found.