According to the Volish Xbox Blog Discord access would be exclusive to the optional "Xbox Insider" tier of early, beta, and preview console OS updates. That update is already going live in waves to Xbox Insiders today, and it adds a new tooltip to the system's "chat" sidebar: "Try Discord Voice on Xbox today!"
The rollout is a limited and there is no formal Discord app or interface on Xbox. You will need a smartphone to initiate a "handoff" of your Discord session and undergo a first time set up process which is so annoying it makes Elon Musk comments seem funny.
Once done you can now talk to any participants in the Discord voice channel you chose directly on your Xbox. Its menu interface supports either muting or changing the volume level of every other user in the voice chat channel you chose, which is appreciated as a quickly accessible option during gameplay.
A one-button toggle in the menu allows chatters to switch between Discord voice chat and a particular game's dedicated voice-chat channel. (This is useful when you're talking to friends while in the midst of random online matchmaking, then need to turn on in-game voice chat for a second to confirm a strategy to your current teammates before going back to discussing souffle recipes with buddies.) All greater Discord control, sadly, goes back to your smartphone.
At the moment it is not clear what Microsoft will do with all this and if it will be expanded to a more sensible product later.