Quite a few reasons, most of them having to do with changing military doctrines and technology. A military doctrine is essentially a playbook, it's a way to ensure that all elements of a military force make the same or similar choices when faced with similar situations, which eases coordination across the board.

Doctrinal changes can come about because of tactical innovations, strategic concerns, the changing role of a military force (such as a force changing from training and preparation for large-scale wars against comparably powerful nations to occupation or pacification, that kind of thing), even politics, trends in training and education, and budget concerns.

I personally follow a school of thought in military history that views technological determinism with suspicion. In other words, I tend to view changes in technology as only a small part of larger trends that express themselves in complex ways that can't be understood solely as incremental technological improvement and its knock-on effects. So before I go on to talk about the effects of a couple of changing technologies and their manifestation in combat, you should just be aware that this is, in my opinion, a rare example of technological changes that had rapid and inestimable effect on militaries around the world. All that said, changes in culture are also responsible for changes in uniform design and aesthetics, but to keep this answer tight I'm going to talk about two things: smokeless powder, and aircraft.

Smokeless powder

Black powder was used for hundreds of years in battlefields all around the world. It was relatively simple to produce, relatively stable and transportable, and widely useful even outside its utility in firing projectiles. It was also somewhat crude, and dirty. One of its side effects is the production of a huge plume of white smoke at every discharge. You can imagine, then, hundreds or thousands or tens of thousands of men all firing black powder muskets, and the smoky haze that would permeate the battlefield.

Battles are already confusing and chaotic, and adding in the increasing obscurement of muskets and cannons and exploding shells makes it all the worse. It takes one perennial problem of military organization - friendly fire - and increases its likelihood significantly. One way to help curb this possibility is to clothe your men in instantly recognizable uniforms. Bright colors and striking hats make the silhouette of a soldier obvious even from a distance, or in the dark, or when obscured by powder fog. This is also, incidentally, why it was so common to carry company, regimental, battalion, or brigade "colors," the flags that mark your unit's location and visibly display some of your unit's history. I've written about musicians and colors in more depth in this older answer.

The introduction of smokeless powder, which started to be widely adopted by the early 1890s, eliminated the buildup of powder smog on battlefields, but it also coincided with new generations of powerful, long-range rifles. These rifles were themselves the product of changing doctrines and changing thought about the efficiency of projectiles which moved military cartridges from (broadly) large, slow-moving bullets to small, fast-moving bullets. Military forces were increasingly relying on skirmish-type tactics, moving fast and low from cover or concealment to cover or concealment. I bring this up mostly to say again that even though this answer concentrates on a couple of pieces of technology, these changes didn't happen in a vacuum and weren't adopted for reasons that are free of politics, fashion, or (for lack of a better term) lobbying.

Because there was less powder fog on battlefields, and because men tended to fight more spread out, no longer marching in elbow-to-elbow formations and massing fire, the need for that instantly recognizable silhouette was less pressing. It never entirely went away (and is still of great concern to modern militaries), but the need for a mass of color-coordinated men moving into the fringes of a fog of powder smoke to fire a massed volley at an enemy was no longer the dominant doctrine. New doctrines stressed rapid assaults, using cover and concealment to advance to as close to the location you want as possible, and then massing for a charge, or engaging with fire from the nearest available cover. In the black powder mode of warfare, this was an inefficient way to take ground. Individual soldiers' individual fire lacked any practical or psychological assertiveness, and so massed fire or massed bayonet charges were used to take ground. With repeating rifles and smokeless powder, skirmish style tactics could take ground. Over the course of the next couple of decades, doctrines changed from preserving fire for its maximum effect - firing ordered volleys at relatively close range and discouraging rapid fire even with repeaters - to using a superiority of fire to put the enemy on their back foot. This changed really didn't occur until well after the introduction and proliferation of the machine gun, though; rifles alone weren't meant to accomplish this.

Aircraft

Imagine you're a general. Your job is to protect your country's capital city against an enemy invasion. You know that the enemy force is somewhere to your north, coming down one (or more) of several wide and well-maintained roads that have subsidiary roads, farm roads and market roads and the like. You know the enemy force is roughly (making this up) 40,000 men, but you're unsure of their exact immediate objective. They might try to take a large city to the north of the capital, or they might make a straight run to the capitol, bypassing strongpoints and looking to force a large-scale, decisive battle.

So, how do you figure out what they're up to? Well, you send guys to go look. Problem is, in between you and the enemy general and all the info you want are enemy guys who also want to count your guys, see what type of guns they carry, capture prisoners and see what they have to say, assess the logistics and supply of your army, how many guns you have, etc. These are skirmishers, scouts. They literally have to go look at stuff, and then bring it back with them to tell you about it, or send a trustworthy messenger. But they also have to fight or avoid enemy scouts there and back. It takes expert scouts to get reliable information. It's a terrifically hard, violent, uncomfortable job, and you can count each piece of information you get in dead scouts.

It's hard to underestimate how much aerial scouting changed the math. A plane can just go look. it can just fly right over top of the enemy's best, most ruthless, most effective scouting screens. It can mark with a pencil every supply dump, every artillery park, every route of march, every column of soldiers, every bivouac. It can do this in hours, what formerly took days, at least.

But the important change is that now there's just a guy, up there in the clouds with a map, using their eyes and just taking information for free. So, apart from getting your own guys in your own planes to not only scout, but also (after some very clever thinker put a gun on one of these planes) screen your army from their guys, you still have to worry about easy information getting to the enemy. So, camouflage. Obscure your supply dumps, hide your artillery parks, try to obfuscate your strong points. And, importantly, try to camouflage your soldiers. Its nothing sophisticated, just take browns or grays or greens instead of the older, brighter colors. Get rid of the standards and colors. Make sure camps are protected or hidden, try to misdirect your enemy with frivolous movements and deceptions.

You can see what I'm getting at, here. The introduction of aircraft changed the very nature of warfare at a fundamental level, just from the ability of aerial scouting, before even discussing the possibilities of aerial bombing.

Now, I'm not exactly saying that changing uniform colors and ostentation is a direct result of this; this is all very complex. Mostly, changing uniform aesthetics have to do with changing ideas, and those ideas are hard to cleanly trace through time. Around the same time that air power and smokeless powder were taking hold in the minds of military experts as reliable tools, there were also other new inventions like machine guns and over-the-horizon artillery, supply trucks and more reliable and widespread rail travel, increasingly sophisticated logistical capabilities, massive increases in the size of armies, national drafts, and popularity and support for violent nationalism. Army doctrines were changing as a result of all of this, and in the first decades of the 20th century we start seeing armies and doctrines that look recognizably modern.

One expression of this spiderweb of changes are in uniforms. Though I would argue that most military uniforms were always designed with functionality in mind - they were often not much different than civilian clothing except for decoration - the mid 20th century saw a de-emphasis on color, ostentation, and pageantry. Those never entirely went away of course, many modern dress uniforms are quite lavish, but the relatively simple camouflage pants, jacket, and cover was an expression of what armies perceived as their need for men in the field. And bright colors, gilded adornments, and tall hats were no longer considered suitable or necessary.