Just a quick note, but not every officer position in the US military requires a degree: warrant officers for instance, do not.

Also, technically, commissioned officers in the US military do not require them to per Title 10. However, each branch prescribes its own requirements to receiving a commission, and in practice, commissioned officers will have had a college degree to simply apply for Officer Candidate School (OCS) or as a byproduct of completing a service academy (US Military Academy, Naval Academy, Air Force Academy, or Coast Guard Academy) or a Reserve Officer Training Corps (ROTC) program.

As far as your question goes:

When did the US military start requiring degrees to become an officer and what was the lead-up to that decision?

It's actually a bit of a reverse in causation in some ways: the US military didn't start requiring degrees to become an officer. Instead, its officers all already had educations that could have only been taught in places of higher education

War in the 18th and 19th centuries was already quite technical. Proper use of artillery, construction of fortifications, siege warfare, etc. all necessitated the use of mathematics, principles of civil engineering, physics, and other specialties which we associate with STEM degrees today.

The US, during the Revolutionary War and the in the years after, had heavily relied upon foreign engineers and military experts. To alleviate this reliance, President Jefferson founded the US Military Academy (West Point) in 1802 to teach the principles and science of warfare. Perhaps not surprisingly, due to their education in the maths and civil engineering, many of the early civil engineering projects regarding lakes, levees, bridges, harbors, etc. in the US were spearheaded by the US Army Corps of Engineers and its officer corps.

As the 18th century moved on, the Morrill Land Grant Act of 1862, and the Morrill Agricultural Act of 1890, both granted lands and funds to the states for universities that specialized in specific focuses related to agricultural, mechanics, AND military tactics. These acts created the "land grant" universities and they had curricula that included military training. Some of these schools - like Virginia Tech and Texas A&M - started off as military schools where all students were cadets, and to this day both of those schools (as well as a few others) still maintain a Corps of Cadets with dormitories and the like.

Remember that prior to the American Civil War, those with formal military educations were few and far in between and the US Military Academy and Naval Academy could only produce so many graduates in a year. In part because of the Civil War's sudden and large demand for officers, especially qualified officers, the idea of a "reservist" group of officers was envisioned and many of these land grant colleges would be able to provide men educated in the sciences and leadership necessary for war that could be called up in times of emergency.

WWI and then WWII would later validate this idea, as well as introduce use of Officer Candidate School programs for graduates. As for West Point and the Naval Academy themselves, their educational focuses put them in line with other colleges and universities and they themselves became accredited in the 1920s, with degrees retroactively conferred to all living graduates in the 1930s, and then all graduates later. Source: Carved from Granite: West Point Since 1902

Even today, the original focuses on engineering and sciences dominate the curricula at the service academies, and taking a certain number of calculus and physics classes are requirements to receive ROTC scholarships.

As thus, the de facto requirement for degrees wasn't so much a sudden decision or act. Instead, it was a combination of warfare requiring higher technical knowledge than was available to general education, land grant colleges already producing college graduates that would serve as reserve officers, and finally the melding of the two as standardization that would lead to the officers requiring degrees be commissioned officers.