press the flesh

What a politician does (or should be doing if s/he hopes to get elected) during the run-up to an election. It means to make contact with the voters by means of personal skin to skin contact (but hopefully non-sexual), such as shaking hands and kissing babies. In a general sense it implies getting out and getting in the face of as many different voters as possible because if they meet someone they're probably more likely to have a positive opinion of them than the candidates that they didn't meet in person.