I think there’s a couple reasons:
1. Tyrion needs a new ruler to place on the Iron Throne in place of his family if he wants to live in Westeros, let alone inherit Casterly Rock.
2. Dany’s got the dragons. She’s the smart candidate to back.
3. They share a hatred for his family.
Regarding the Blackfyres, the key here is to look at family trees. Daeron II (who’s grandmother was a Rogare) married Mariah Martell; Maekar I married Dyana Dayne, Aegon V married Betha Blackwood, etc. Dany’s father and grandfather both married their sisters, but that was in the wake of a lot of out-breeding.
But the Blackfyres have a different descent. Daemon was the son of Aegon IV, but he was also the son of Daena the Defiant, who was the child of Aegon III and Daenaera Velaryon, who have both Valyrian and Targ blood in their lineage. Thus, Daemon had more Targaryen blood than his half-brother. Given Daemon’s claim from both of his parents and his desire to marry his half-sister Daenerys, it’s quite likely that his children practiced more consistent in-breeding than the main Targaryen line – hence Maelys the Monstrous.