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.
Mr. Bush probably doesn't need to go out and press the flesh because he is running for re-election based on his first term record.