I'm actually a part of the Xbox one preview program so I have gotten to play around with this stuff a bit.
Banjo Kazooie and Tooie run flawlessly, as well they should since they're simple arcade titles.
The way it works is, you download said title that you own digitally to your Xbone, boot the game, which will then take you past an Xbox 360 splash screen, followed by authenticating and signing in your Xbox 360 profile. After that, the game will play.
Now something to note. Signing in with the 360 profile will, obviously, sign it out anywhere else. So if you did hold onto your 360 (or have like 6 like I do) and if you happen to have little children that still play 360 games using the account that has live on it (like I do) it will kick them off, which will result in a lot of fighting over who gets to use the profile when.
Otherwise, the disc method also works fantastically. Simply putting in the disc will then have Xbox download some compatible files (pretty much it just downloads the disc, but still needs to check the disc to actually run it, like most Xbox One games today) and then boots up and goes.
It's all very seamless and works fantastically, I'll try to upload some screenshots using the Xbox One overlay soon.
Which is actually another point f interest, is the fact that you can now Screenshot, record, and stream the 360 content THROUGH the Xbox Ones built in apps such as Studio and Twitch.
Very very nice, can't wait to see more added to the ecosystem.