Sometimes it's a case of ideas popping into our heads when we've relaxed our minds from the problem. eg Some of my best ideas come to my while I'm on the toilet because I can be staring at a piece of code for hours and not find the fault. But the act of walking away from the problem and relaxing my brain for a few minutes is sometimes all that's needed to see the problem from another angle (and thus stumble onto the solution).

It's the same reason that I used to have a bath whenever I'd get stuck on a college assignment.