Puzzle Mode is unlocked from within adventure mode, at the same time as unlocking the second area. The second area will unlock once you finish all three adventure puzzles from the first area (Grumble Tummy Grove). So simply start adventure mode and compete the first three levels and you will earn this trophy.
The opening levels are pretty straightforward and slow enough that they can pretty much be beaten simply by feeding critters without aiming for fancy combos or food chains. So if you are having issues completing a level you can focus on just passing the levels. However, if you can, its a good idea to feed Smalls in these levels so that you can make progress towards the 'World's Greatest Dad' trophy.