First off: NO SPOILERS HERE. This is a game where I believe being spoiled can significantly impact your enjoyment, so I'd advise people to avoid them at all costs if they're interested in playing the game.
So I was a bit wary going into this game because of the amount of praise it's gotten in the past as well as its comparisons to such gems as Ace Attorney, Persona 3/Persona 4, Virtue's Last Reward/999 and Battle Royale, but Danganronpa: Trigger Happy Havoc really does incorporate important elements from all of those:
The game's murder investigation scenes are pretty much the same as in Ace Attorney. Walking around the school, gathering evidence by finding evidence and interrogating witnesses to prepare for the "classroom trial." The actual trials have a different dynamic, since there's no prosecution or defense and every student basically takes part in them to determine the suspect, but that lends a different kind of intensity to determining the outcomes of a case since they involve characters you come to know.
Outside of "Nonstop debates", which are essentially sped up versions of cross-examinations in AA, the actual gameplay during the trials is fairly different than Ace Attorney. Some of them are basically like mini games, though they all essentially revolve around presenting contradicting evidence to statements. On this note, I'd recommend anyone who starts playing to set the "Logic" difficulty at the beginning of the game to "Mean." I went with the default "Kind", but that seems to remove a lot of the tension when it comes to the classroom trials.
Persona 3 / Persona 4
What Danganronpa has from the modern Persona games is less significant than what it has from Ace Attorney, but DR has moments which are labeled as "Free Time", when the story isn't actively taking place. These are moments during the game where you can freely roam around the school and interact with fellow students, which will take time out of the day.
Interacting with students will get you to know more about who they are and, as the main character spends more time with them, their relationship will grow stronger. Growing relationships with characters will unlock new skills or abilities that will come in use during the courtroom trials.
999 / Virtue's Last Reward
What Danganronpa shares with the Zero Escape Series is its thematic. There are no time traveling or morphogenetic field shenanigans here, but Danganronpa deals very largely with the sense of "despair", how one's appearance can be deceptive, whether to trust someone or not, the "prisoner's dilemma", etc. These students are stuck in a location against their will, by an unknown mastermind, forced to either cooperate or die painfully.
The element added by Battle Royale is that how the game is initially set up, only one can survive while the rest will die, and no amount of cooperation will change that.
The game has a lot of black comedy but, when it gets serious, it gets serious. I actually wasn't expecting it to be as dark as it is coming into the game.
I'm only almost done with chapter 1 after finishing my first courtroom trial, but I can tell that this will be a lot of fun. If you enjoy the Zero Escape series, at least, I'd recommend checking it out.