How or when did baseball's 7th-inning stretch originate?

I think it has to do with “last call”.

President William Taft was at a ball game. He got up to stretch his legs in the 7th inning. People thought he got up to leave, so they all stood up.
This was back in the time when people still showed respect for their president.

There are other theories, but I like this one best.

The other tales from Wikipedia

“One claimant is Brother Jasper of Mary, F.S.C., the man credited with bringing baseball to Manhattan College in the late 1800s. Being the Prefect of Discipline as well as the coach of the team, it fell to Brother Jasper to supervise the student fans at every home game. On one particularly hot and muggy day in 1882, during the seventh inning against a semi-pro team called the Metropolitans, the Prefect noticed his charges becoming restless. To break the tension, he called a time-out in the game and instructed everyone in the bleachers to stand up and unwind. It worked so well he began calling for a seventh-inning rest period at every game. The Manhattan College custom spread to the major leagues after the New York Giants were charmed by it at an exhibition game, and the rest is history.[1]

However, a letter written by Harry Wright of the Cincinnati Red Stockings dated 1869 – 13 years earlier than Brother Jasper’s inspired time-out — documented something very similar to a seventh-inning stretch. In the letter, he makes the following observation about the fans’ ballpark behavior: “The spectators all arise between halves of the seventh inning, extend their legs and arms and sometimes walk about. In so doing they enjoy the relief afforded by relaxation from a long posture upon hard benches.” Another tale holds that the stretch was invented by a manager stalling for time to warm up a relief pitcher.”

@charliecompany34

We Cubs fans embrace the Stretch because of our beloved Harry Caray, but we can’t forget that he sang at Comiskey long before heading to Wrigley.

@Seek_Kolinahr I figured Cubs fans just got up and left after 7½. I mean, why stay?

@filmfann

Ah, that’s only during the Playoffs. During the season, we always stay to watch them mop the floor with the other guys.

It was just after the first few “6th inning nap” occurrences, and was developed as an emergency measure to resolve the “8th inning ruptured bladders”.

lol @Seek_Kolinahr‘s “other guys”