Shuffle Time is a random mechanic that can activate after every battle.

How do you activate it, you ask? It's simple. It can activate randomly after a battle, but these are the most consistent ways of activating it:

Killing the enemy with their weakness
Killing them with a combination attack
Killing them with an all out attack
Killing them with follow up attacks

Sweep bonus is something that happens when you draw every card in a shuffle time. When you trigger a sweep bonus, you can draw more cards the next time you do shuffle time. This lets you increase your stats in shuffle time a lot easier.

The best way to trigger a sweep bonus is to draw the cards that let you draw more cards. Usually these cards will have something like "1 more" or "2 more" on top of them. Draw those cards first and then draw the others to trigger a sweep bonus. After you trigger a sweep bonus, you're almost guaranteed to have another shuffle time at the end of your next battle. You can use this to your advantage to increase your stats, money, experience, skill cards, and also get new Personas to add to the compendium to make progress towards the Fusion Expert, Special Fusion Expert, and the Thorough Bookkeeper achievements

I hope this helps.