Picarats are essentially "points" that you receive in exchange for completing puzzles. The harder the puzzle is, the more Picarats you can receive for solving it. When you enter the wrong answer on a puzzle, the amount of picarats you can get decreases by at least 10% (for example: if the puzzle is worth 30 picarats, you will lose at least 3 picarats for a wrong answer). This will only happen the first 2 times, after that they will no longer decrease.
Picarats, despite seeming trivial and unimportant, are actually necessary to unlock content in the Top Secret section of the Bonuses feature of the game, such as character profiles and game music. This encourages the player to try and solve as many puzzles as they can, and are encouraged to really think them through and get the answer right on the first try.
In Professor Layton vs. Ace Attorney, picarats are awarded at the end of each trial. The amount depends on how many penalties you received during the trial.
"Picarats are units that tell you how difficult a puzzle is. When you solve a puzzle, you will collect the picarat value of that puzzle. Collect as many picarats as you can to unlock extra content in the Top Secret section in Bonuses."
"Picarats are units that tell you how difficult a puzzle is. When you solve a puzzle, you receive the number of picarats that was displayed on the introductory screen. Collect as many picarats as you can to boost your total and unlock extra content in the Top Secret section in Bonuses"
"Picarats are a unit of value that indicate the difficulty of a puzzle. Collect enough picarats, and I do believe you might be rewarded for your efforts!"
"Picarats, dear reader, are a unit of value that shows the difficulty of a puzzle. When you collect enough picarats, I’ve heard delightful things can happen."
"Well, you see, picarats are units that measure the difficulty of a puzzle. Rumor has it wonderful things can happen when you collect enough picarats."