I guess a real, real answer would be
I find fun in competing. I feel like fighting games break it down to a simple concept of fighting myself against my opponent. It's a combination of reaction timing, reading people, and deploying techniques into a fight.
The reading people is a huge thing and I find a lot of fun in it and manipulating people into thinking you are going to do one thing and instead do another is a surprise and I like surprises. I also like comebacks and with fighting games this is something that is more realistic than say StarCraft or DOTA where a comeback is likely more of a result of the opponent's mistakes than getting gud.
I like that fighting games isolate to themselves, too. It feels like a community. I find Smash Brothers to not be in this community persay (due to a lot of factors), but I think they have a very similar thing going for them and would consider them at least cousin communities.
DOTA-style games do a thing where a character has a lot of lore to follow and you can play the character and win. I feel that fighting games share this, too. This grabs me, too. I tend to play characters that I like personality wise or story wise.
For example, I play Ken basically because I like his smile.
With Phoenix Wright, for a more real answer, I liked that he was the worst character and that he was a defense attorney. It made turnabouts very fun. There was a certain hype there that just got me excited to bring him out against people who I never had fought before.
And that's really what it's all about in the end. Testing yourself and your opponents. Finding someone you are 50/50 with, is a really neat experience. Get yourself in a room with four others of similar skill level, and there is a really good time to be had. When someone deploys new tech, it will destroy. People will then try to have to counter that tech with other tech and it becomes a constant arms race of new ideas and strategies.
This is the same with all games, but a match only lasts less than five minutes and it's a fight. It's hype. It's aggressive. It's competitive.