The contractor repainting my house told me that, to find potential problems with the house, you look at the siding and "you've got to let it talk to you". You stare at the shingles and "they talk to you": chipped corner, darker stain, soft wood, insect holes, faded colors. They're all symptoms of a variety of problems. You see the problems, put two and two together, and figure out where they come from.
What he described was an approach to problem-solving that can readily be applied to mathematical problems. You look at a series of examples. You stare at them attentively, you let them "talk to you", and the structure of the problem comes through.