Why do people wear sunglasses in The Matrix?

For the rebels, it's same reason they wear long trench coats and mostly black clothes. It's a stylistic choice, that both sells the rebels image, that they fly against convention, and the cyber punk look from futuristic stories like Blade Runner and Ghost in the Shell.

When Morpheus meets Neo, he is wearing sunglasses, despite being inside a house during a rainy night. It's just meant to look cool. They'll shed the shades and trench coats when they need to be more practical.

Trinity in particular also uses her sunglasses as eye protection when riding a motorcycle twice, both in the first and second movie.

For the agents, it's another deal altogether. They wear it as a disguise. An additional layer of impersonality. It's standard MIB fantasy clothing. A dark suit and shades.

To the best of my knowledge, the Wachowskis never commented on the shades in particular, but I'll leave here a nugget about shades:

Psychologist Paul Ekman, an authority on facial expressions, says that of the 23 facial expressions relating to human emotion, about one-third involve the eyes. Shades may slightly obscure your view of the world, but they also hide the world's view of how you are feeling.

The sunglasses for the main characters were designed by Richard Walker from Blinde Design. In this article in the Chicago Tribune Kevin Jones (curator of the museum at the Fashion Institute of Design and Merchandising in Los Angeles) talks about design choices in the production design:

"What was interesting about those in 'The Matrix' is the way they reflected back to the viewer what they couldn't see on screen," said Kevin Jones, curator of the museum at the Fashion Institute of Design and Merchandising in Los Angeles. "In just about any film, when you see a person driving a car, they take the rearview mirror away. But here are these sunglasses acting as a rearview mirror so you know what's going on behind you as well as in front of you."

Conversely, there's the element of mystery.

"You cannot see where the eyes are looking," Jones said, "and that can be exciting to the plot."

So out-of-universe it was definitely a specific choice for the production team to get a unique, individual design specific to each character.

An interesting in-universe interpretation can be found here:

The renegades and the Agents always wear sunglasses in the Matrix. Sunglasses hide the eyes and reflect those who are being looked at. The removal of sunglasses signals that a character is gaining a new or different perspective, or that he or she is vulnerable or exposed in some way. When Neo removes his glasses to kiss Persephone in The Matrix Reloaded, he looks deeply into her eyes, indicating both the precariousness and gravity of the moment. When Morpheus offers Neo his crucial choice between the pills, the blue pill is reflected in one shade of his sunglasses, the red pill in the other, an overt reference to the two different ways of seeing that Neo must choose between. When Neo enters his new world, his sunglasses serve as protection for him, keeping him invulnerable to the dangers and surprises he encounters.

One final interesting point - at the start of the film all of the agents' sunglasses are the same shape, but as Smith goes increasingly rogue, his sunglasses begin to take on a new shape (as noted in TV Tropes):

Special mention goes to Morpheus' reflective pince-nez shades and the change in Agent Smith's lenses. They start out with the same oblong shape as those used by other Agents, but once he goes rogue, they take on a polygonal shape that approximates the outline of Neo's shades to contrast their growth.

In the related video game Enter The Matrix, it is implied that the sunglasses worn while in the Matrix are actually a heads-up display. You see a typed message from Sparx appearing character by character, superimposed on your view of the game.

EDIT: I should emphasize that there is zero support for this idea in the movies, and it probably was invented by the game developer. Nevertheless the Wachowskis probably signed off on it.

Lots of interesting answers, including the bullet points from the research you have already done. I like almost all of them.

However I didn't see one that clearly gives the in-movie explanation. All of these answers are very good reasons to why the choice was made to have them wearing glasses, but in movie, its much simpler than that.

Because they choose to.

In Morpheus's explanation of what the matrix is to Neo, he points out Neo's appearance as a representation of himself. The way he sees himself is reflected in that. Do they think the glasses make them cooler, so they have them because they think they are cool? Maybe. It doesn't matter. Thats how they view who they are.

Im glad they view themselves with such awesome eye ware, as the scenes that you can see the reflections would be very different and much duller without them.

They wear sunglasses because making animations in sunglasses is easy and the animated scenes in sunglasses look more realistic. Considering the animation technology we had in 1999 when movie came, it looks a fair choice.

For an in universe answer for a spy sunglasses would hide their eyes so others cannot see their expression as easily or where they are looking. Poker players wear them for this reason.

An out of universe answer is given in this video.