In the military sense, the phrase campaign means “military operations for a specified objective”. In the old days, it meant military operations for one season. Back when you didn’t fight in the winter each late spring into summer was “campaign season”. It would eventually enter into wider usage as a political term, but still kind of hold s the “operations for a specific objective” meaning.
The term came into usage in the mid-1600’s and took its origin from the Late Latin “campania” meaning level ground. This is where “camp” (field) came from. When an army left its fortifications to start an operation it literally, “took the field”. And thus went on campaign.
This actually can tie into the political meaning of the word also. See up until recently, at least in the US, when someone ran for office they relied on others to go out and drum up support. So in effect, they were sent out to take the field, or campaign or their candidate. Huh, sort of makes sense.