Persona 5 has multiple endings, but Royal adds a new ending. To get to the third semester (aka royal’s content) you need to have 3 confidants reach specific ranks.
Guide to Royal Ending
You need to reach Rank 9 with Maruki and Rank 8 with Akechi before the 18th of November, and reach rank 5 with Kasumi before the 18th of December.
After you reach these ranks, you have to make specific dialogue options to continue to the third semester. Very minor spoiler warning below (Does not spoil what happens, but does tell you the date they happen on).
You’ll want to play dumb, select “No” or “Never heard of them” and then “There were no such people” on the question after.
“I refuse” is the answer that will take you into Royal’s third semester.
You’ll want to refuse again. This is the last of the major choices.