The Episode Bonuses are unlocked upon completing all 6 Chapters/Levels in each respective Episode. There are two Gold Bricks to collect from each set of Bonuses and three challenges to complete in total. They are as follows:
Super Story: Play through all 6 Levels over again in Story Mode, playing the whole Episode in one sitting. Unlike LSWII, you don't have to beat the best time (1 Hour) or collect 100,000 studs to get the percentage. Simply beating all the levels in one sitting will be all you need to do. No Gold Bricks for these, but you do need to finish them for 100%. I will provide tips for beating the best time and best score but know that unlike LSWII, you don't need to do this.
Character Bonus: Pick two characters and you'll be deposited into an arena based on the Episode you're currently in. Your goal, collect 1 Million Studs to complete the mission. Like LSWII, the elusive Purple Stud appears in both these levels and the ones below, giving 10,000 studs. Power-ups are very useful here due to the x2 Multiplier they give.
Minikit Bonus: Unlike Character Bonuses, these won't unlock until a Minikit Model has been completed. Once done though, the levels open and they're available. They're very similar to the above Character Bonuses but for Vehicles instead. Collect 1 Million Studs to complete the level. Generally easier than the Character counterparts.
Note that you can use Extras in the Character and Minikit Bonuses. Anything regarding Studs and Money is disabled, but everything else is fair game! However, you cannot use Extras in Super Story.