Tricky question to answer, but I'll take a stab at it. Just took a military history class over the summer and the idea I took away from it is that at the start of the US, we had the Second Amendment because of the way society was structured. There was no standing army like we have today, so instead we had to rely on citizen-soldiers and militia. In addition, at the time, you've got settlements that have to deal with Natives and raids and the like, meaning it was basically essential to have people with weapons. Alongside the lack of a standing army, there wasn't much in terms of a police force, meaning that when raids occurred, people were on their own. [Millett]

The lack of a standing army also plays into this: looking through time at wars, we had an 'accordion effect' where the army would expand during war time and contract after. So, for the Civil War, the army expands to over a million men, but is back to 11-12,000 men by about a month after the war. [Millett]

Prior to this, Americans weren't ever stoked to pay taxes for a standing army. In the Spanish-American War, you can still see at least remnants of the militia approach in Theodore Roosevelt's Rough Riders. This would have continued into WWI, as Teddy tried to round up a group of Rough Riders, but for political reasons, Woodrow Wilson intervened to prevent this. (Teddy knew that presidents in the pantheon of 'greatest' had led the country through a war, and this was what he desired with WWI; he also probably thought it would catapult him back into the White House). [Brands]

The trends of an accordion-like army and the citizen-soldier ideal continued up to World War II, at which point, a variety of factors (but to simplify, primarily the Cold War with the Soviets that cropped up after) led to the necessity for a standing army. [Millett]

In my opinion, this is when the change turned from the Second Amendment being more about 'militia' to being more about 'rights.' I'm sure there are a variety of reasons for this, among them likely the same fears that the Founding Fathers had about a standing army: James Madison refers to a standing army as an "instrument of tyranny." Part of it likely had to do with people who had grown up used to guns and began to consider it a right, similar to today, but at this point, I'm stepping into speculation. [Millett]

Sources: For the Common Defense, by Alan Reed Millett Traitor to His Class by H.W. Brands