Mean Bean mode is unlocked by getting gold (or silver) medallions - the exact number is unclear. You can see my guide for unlocking those for the following achievement:
The boss of Chemical Plant Act 2 is also this minigame, so you can earn it in either version of the game.
The minigame is actually
Puyo Puyo Tetris (or, well, the Puyo Puyo part of it). It was also a classic 16bit (or 8bit?) game. This is a very popular game in Japan that's slowly gaining some cult popularity here. It's very close to Tetris. You'll get colored blobs in groups of two and need to match them in a group of four to eliminate them. It's pretty simple once you see it in action.
A 3 chain combo is where you pop 4, then another 4, then another 4 all in one action. This isn't incredibly hard to set up, but how you do it depends on what pieces fall. You can see me earning it in the video below: