What happens with these puzzles is that your brain gets locked into the initial configuration and assumes the sticks are already in their "correct" form. But really, all you need is to allow yourself to see those same sticks in a different way. A stick that's now part of the number 5 can become part of the 3 or the 8 if you move it a centimeter. The equation that seems impossible suddenly makes sense when one of the numbers transforms into something completely different.
The other factor is that these puzzles almost always play with Roman numerals or the specific way we draw digits. For example, if you see something like VI + V = XI and they tell you to move just one stick, the tendency is to think "VI is 6, V is 5, that adds up to 11, so it's correct." But it could be that one of those sticks is part of two numbers at the same time, or that moving it transforms one number into another that makes the equation work. I spent a while staring at one where you had to move a stick in the number 1 to make the whole operation make sense, and it turned out that stick could go to form part of the plus sign or the next number, changing the whole visual interpretation.
The logic is actually pretty simple once you see the solution: it's basic math, but wrapped in a visual trick your mind doesn't expect. The challenge isn't in solving complicated equations, but in training your eye to see that those sticks aren't the "definitive numbers," just one of many possible ways to represent digits.