Some of the currently public achievements are tied to story events, such as completing campaigns and the raid. Others, however, clearly involve pouring hours upon hours into other events, such as reaching Vanguard Level 50 or the Legend rank in Valor (a long-term ranking for Crucible play). Much like in the first Destiny, some of these Triumphs will be difficult to pull off unless you've been playing for a long time.
The rewards will be relatively familiar. You'll largely receive in-game gear tied to the Solstice of Heroes summer event. If you rack up at least 250 Triumph points out of 400, though, you'll receive a code that lets you buy a customizable t-shirt. Yes, it's a bit silly to grind away for the 'privilege' of handing over money, but it's one of those rare instances where your in-game skill can translate to a tangible memento.
You can check out your character's profile on Bungie.net right now to see your progress, although you won't get to claim any rewards until the Solstice of Heroes event starts in earnest on July 31st. Just be ready to hustle if you find yourself lagging -- you only have until August 28th (conveniently, a few days ahead of Forsaken's release) to earn what you can.
The Moments of Triumph release isn't entirely unexpected. Bungie has been scrambling to offer reasons to come back to D2 in between story updates, and that was bound to be a challenge with four months between Warmind (the last of the year-one add-ons) and Forsaken. The use of Moments of Triumph relatively early into the game's lifecycle may help with that, especially for players who might have been regulars early on but ran out of things to do.