I played Persona 3 and Persona 4, Persona 5 should be similar so here's how it goes.
You are a senior year Japanese student in high school. The game has two 'sides', the morning 'social' side where you attend school, talk to your teammates, go to the mall, help people, etc. The more 'friendly' you are with a teammate, you get extra combat bonuses, also you get a partner-specific ending. Basically you live the student life for one year.
The night 'dungeon crawler' side is when you go down to dungeons, to kill some boss mob or whatever the goal is. To do that, you capture different 'personas' with different abilities and stats that you use in combat. For example, the 'Pegasus' persona has air attacks and is fast, while the 'Cerberus' persona has fire attacks and good dodge, and the Golem persona has great defense, earth attacks but is slow (this is all made up as an example). These personas grow up in levels to learn new abilities and increase stats, but eventually you want to transform those personas into new ones (inheriting some of the abilities the previous persona had) so there's a strategic side to Persona management.
Combat is turn based and tactical because of the Persona management thing. Also that's how you earn new personas which you can combine with other personas to create better ones.
In my case, one of the things I like the most about these games is that, as a Japanese student, you get to experience and learn about Japan's rich culture. Other people love the art style, and/or the music. So there's something for everyone.
