I know I know, you want to know what the mystery fruit is. I didn't mean to leave you hanging.
This is a quenepa, which is called a million different things in different places. It's a tropical fruit which looks a little like a lychee here, but what looks like a luscious solid fruit is actually a thin coating of fruit pulp around a thick seed that can be roasted like a chestnut (though I haven't done that). You peel the fruit open (easily done, just like with lychees and similar fruit), pop it into your mouth, suck the pulp off, and spit the seed out. Though there's very little pulp in the fruit, it's so flavorful that you don't notice -- it's as sweet and tart and intense as passion fruit, with a sort of "tropical punch" flavor. I have, naturally, added a bunch of it to vodka to make liqueur.
As for this mystery?
Dates go through various stages of ripeness, and at the khalal stage, they're crisp like apples -- though tarter and noticeably astringent. Tasty, but you're not going to eat a bowl of them on their own. These are specifically Mariana dates in the khalal stage. They continue to ripen on their own, hitting the rutab stage, at which point they become so soft they're almost liquid, and melt in your mouth, tasting a lot like vanilla caramels. I assume if I leave them long enough, they'll become the dry dates you're familiar with from the supermarket.
As for why I bought fresh dates, well. That's what I do. That's how I learn to cook.