Bravely Default 2 features New Game Plus that allows you to carry over many things from your finished save file. In order to unlock NG+, however, you will need to finish the true ending. Once you finish the true ending, you will be able to start a New Game Plus.
In order to reach the true ending, you will need to first witness 2 bad endings. The first two endings are gotten by just playing the story normally. Experiencing the first bad ending will lead to Chapter 6, and will also unlock Trials of Tribulations, which lets you level up jobs to 15.
Once you defeat the final boss the second time and witness the second bad ending and reload the save file again, the story will hint you towards the secret Asterisk. When this is acquired, you will notice that when you open the save menu, there will be an eerie noise and a creepy purple save slot at the bottom.
If you try to overwrite this save file, you will trigger Chapter 7. Simply follow the story and you will reach the true final boss battle. Defeating the boss this time will allow you to create a new save file, from which you can start a New Game Plus.
As for what carries over in New Game Plus for Bravely Default 2, you can carry over pretty much everything that has an impact on gameplay, plus the bestiary and memories you unlocked. Keep in mind that while Buns in your inventory carry over, the permanent stat boosts you get from them do not.
The difference between Captured Monsters and Number of Monsters Captured is that the first one carries over the monsters you have available to you that you can unleash using the Off the Leash or Off the Chain abilities.
The Number of Monsters Captured is the amount you have successfully captured in total and is used for the purpose of the Beastmaster’s second Speciality, which gives it a stat boost based on the number you have captured.
Carrying over Playtime and Battles won are for the purpose of another job since the secret job has abilities that get stronger the more Playtime and Battles Won you have.
It is recommended to try farming some of the Trial of Tribulations weapons in the endgame before carrying it over to NG+, as a majority of them provide all the job-specific passives when equipped.
An interesting aspect about New Game Plus in Bravely Default 2 is that it allows you to easily defeat Adam in the prologue when the story requires you to be defeated by him. Defeating Adam will result in a “secret” ending, which is rather sinister. The game will also straight up tell you that you’re supposed to lose to Adam and put you back to before the battle started.
There is also something similar for another “unwinnable battle” later in the story, which takes place in Wiswald.