Are you familiar with Xbox 360 achievements? PlayStation trophies are pretty much the same thing. You are given a number of tasks to complete (usually 12 on a downloadable game such as Lemmings), and the reward is a trophy. Your PlayStation Network profile will be given "experience points" whenever you win a trophy (the better the trophy, the higher the experience), and you can level up. As far as I know, you can only use these trophies and levels for bragging rights (unless there's stuff you can do in Home, but I haven't installed that game). Both Xbox 360 achievements and PlayStation 3 trophies are designed to increase replay value.
And yep, you need a PlayStation 3 because the software is designed to run on that system. It's different to the Lemmings on PS2 and PSP. It has new levels, vertical levels as well as the standard horizontal, and skills you can pick up (like Lemmings 3, except collecting the skill will add it to your repository). The one thing this game is missing that would make it awesome is a level editor.