We have written instructions for producing a human skeleton from a corpse going back to 1543, which included specific steps for removing the flesh and leaving the ligaments intact. As you can imagine, it's a gruesome read. But the art of skeleton-making didn't become popular until the 17th century, when medical schools started using them to teach anatomy.
Soon students of both art and anatomy were expected to study human skeletons as part of their training, and the public grew curious as well. By the 1660s, there was a market for them in Europe. By the 18th century, displaying human skeletons became trendy. Guerrini found a 1716 advertisement for “The Moving Skeleton,” a public attraction “which by a mechanical projection performs several very strange and surprising actions, also groans like a dying person, smoaks[sic] a Pipe of Tobacco, and blows the Candle out, as naturally as if alive.”
By this time, anatomists wanted to produce clean, white bones. One physician made sure to leave his bones out for months to bleach in the sun. Another eschewed boiling bones and instead left corpses to rot in water, changed periodically. This “maceration” technique required pulling softened flesh away from the bones and would have required a steely constitution. But the demand for skeletons was high enough that more people were taking on this job: In the early 18th century, one surgeon offered a course in skeleton-making.
Atlas Obscura has more on the history of making skeletons.