This trophy is both time-consuming and annoying. In order to 'Star' a level, you have to earn all prizes (picking up prize bubbles to earn certain scores), collect all presents and place first in each event. There are 71 levels in Story mode you have to do this for.
All levels can be completed on Casual difficulty.To save yourself many, many hours and retries of each level, I would highly recommend buying the 'Timesavers' DLC add-on as it will unlock 100% of all level content in the game for you, meaning you only have to worry about placing first in each event.
The DLC only costs a couple of dollars/pounds and is definitely worth the purchase. Since this is the method I used in order to unlock this trophy, I can't provide tips for each event and/or mini-game - I can, however, tell you that this is considered no easy feat to unlock without the DLC.
Good luck.