So Apollo as a result of losing a war game lost his assets, his familia, and was exiled IIRC. Why did he go through to this for Bell Cranel? Why did Apollo want Bell Cranel for his familia?
Based solely on what was shown in the anime, the suggestion that comes from the banquet in episode 1 of DanMachi II is that:
- Back when the gods still lived up in their own world, Apollo tried to woo Hestia and she turned him down.
- Apollo says he is definitely not still holding a grudge about being turned down.
- Everyone knows that Apollo holds grudges forever, and tends to take disproportionate retribution against those who did him wrong.
From that, the assumption is that Apollo is still annoyed that Hestia didn't go out with him, and in response he's waited all this time for Hestia to have something that she loves dearly, so that he can take it from her.