Was there a backstory or karmic reason mentioned for why the tiger ended up at Sai's feet?

Yes, Chapter 31 speculates on the tiger's past karma. The text suggests the tiger was likely a meritorious person in a past life who committed the sin of insulting a devotee of Hari. This act led to a curse that resulted in its 'cruel birth' as a tiger. However, it seems a counter-curse was also in effect, which ultimately guided the creature to Sai Baba. Through Sai's divine gaze, or 'darshan,' the tiger's past sins were burned away, the bonds of its negative karma were broken, and it was liberated from its suffering, thus achieving salvation.