The most common answer I see is something along the lines of "it's the equivalent of liking a post on twitter". It seems that this is not the case, as the Mastodon devs seem rather adamant that they don't want "likes" in Mastodon. Perhaps it's a method of saving posts? Well, that doesn't make sense either, since there is already the ability to "Bookmark" a post to save it.
It really just seems like a "Favorite" is just a bookmark that tells the poster, and the public that you bookmarked the post. And even if this was the reasoning -- which is baffling enough as it is -- it wouldn't make sense since the whole point of boosting something is to tell the public that you like a post.
It really seems like the "Favorite" button has no actual unique purpose. In my honest opinion, Mastodon should just federate "Likes" like normal, and be done with it.
The official purpose and the common purpose are not always in sync. That same style of question had been discussed with regards to the lemmy/reddit downvote. Officially, it de-prioritizes the post/comment in sorting due to being disruptive or unhelpful, but the common use is more tyically 'I don't agree with this statement'.