Biomutant has plenty of puzzles throughout the course of the game, but they all tend to fall into the same type. You need to move some switches to match up some lights and only then can you progress. What makes them challenging is that they tend to have a limited number of moves that you can try.
As such, the real trick to these puzzles is to figure out how many moves you need to make before you start actually moving the dials. Examine each puzzle, noting the colors that you need to match up, and where they are on the “dial”. Each stage will normally have two colors in opposing positions, and the dials will often start with one color being closer to the their destination, indicating which way you should start moving first.
Hitting the left trigger will move a dial anti-clockwise, while hitting the right trigger will move the dial clockwise. You can switch between dials without costing yourself any moves using the left analog stick. The aim of the game is to line up all the colors next to lights of the same color, and this will always be achievable within the 10 moves that you will be limited to.