To travel around the country or state making political appearances during a political campaign. The phrase was first used when pilots would travel around the country to entertain with their flying skills.
All-Things-Aviation: “A typical barnstormer (or a group of barnstormers) would travel across to a village, borrow a field from a farmer for the day and advertise their presence in the town by flying several low passes over it – roaring over the main street at full throttle. The appearance of the barnstormers was akin to a national holiday. Entire towns were shut down and people would flock to the fields purchasing tickets for the show and plane rides. Locals, most of them never having seen planes before, would be thrilled by the experience. In several towns parties would be organized on such occasions in the honor of the barnstormers.” Now the word has come to mean a political speech given on the road.”