I think you might consider asking instead, why do we not wear hats very much? From medieval iconography through 19th century photos, there's quite a lot of hat-wearing, even inside. Various reasons exist, beyond fashion accessorizing ; more people living more of their lives outside, people needing to be warm ( notice the number of portraits of Tudors with fur-trimmed robes), the lack of sunglasses, the preference for white, un-tanned skin as a sign of beauty.

To sum it up, for cleanliness. Keeping one's head covered keeps the hair cleaner and neater. Not only avoiding outside issues like smoke (fires year round for cooking), dust, and dirt, but it also helps to keep the hair less oily and less tangled. Having the hair down, unstyled, and unkept is usually reserved for children through most of history. It also makes having a bad hair day impossible since it's so easy to throw it into a cap.

There is also a religious/conventional aspect. Crusaders brought the concept of women covering their heads and hair to be pious, modest, and respectable back from the East in the high Middle Ages. And according to A Brief History of the Tudor Age, men of the Tudor period had similar rules about hats as more recent society regarding removing them or keeping them on to show respect/status.

At the same time, the Little Ice Age began around 1350 - it was noticeably colder then, and while "my head is cold" is rarely the kind of thing that influences society consciously to this level, it's very likely that it helped create a subconscious bias in favor of wearing hats (as well as the numerous layers that remained an important part of fashion for a long time).