"Hey, boss. How do we transition from Monokuma's theme to Phoenix's theme?"
"We don't."
In all seriousness, what's going on with this "King for Another Day" tournament? Is it like a popularity poll or something?
It's a popularity poll in which 32 constestants compete to take over the SiIvagunner channel for a day. The channel mainly focus on videogame parody songs and mashups. Monokuma and Phoenix Wright are participating as one single contestant, called "Law & Disorder".
Each character represents sources for parodies (or "rips", as they call it), and in Law & Disorder's case, they represent mystery games soundtracks, as well as court-related music. The tournament works in a double elimination format, and thoughout the event "rips" like the one above are uploaded to promote the contestants.
There was another tournament like this last year, and Unregistered Hypercam 2 won, giving us some great rips. This year, the number of contestants doubled, and the newcomers were announced on Silvagunner Directs (yep) and in surprise reveals in the end of regular rips, with a cool jingle to grab the attention of the listener and all. There's also this cool subplot where Mr. Bean was the final contestant, but got attacked and Phoenix and Monokuma took their place and are supposed to investigate who did it to clear Maya of suspicion.
That being said, that was the Smash-like announcement that will keep me happy even if those two don't get in the actual Smash, especially since I rooted for them to be announced as a duo since the tournament was revealed.
You can check more about the tournament
here, as well as meet the other characters. There are some great choices, such as MissingNo representing glitched games and creepypastas.
And remember: vote for the sources, not for the characters.