AskHistorians

The red uniforms originated from the Tudor period in British history. The royal bodyguards wore the colors of the House of Tudor (red and gold) specifically because it made them easy to identify. During the Wars of the Three Kingdoms, the English parliamentarian military adopted the colors for their uniforms, and it ended up being the standard for British military dress for the next (nearly) 300 years.

"Sticking out" wasn't as much of a problem back then, because guerilla tactics were less common. That was in part because long-range warfare didn't really exist until much later: even if you had a gun, you'd typically get one shot out and then need to run in with your bayonet versus spending multiple minutes reloading. In a huge melee skirmish between two armies, being able to quickly and easily identify friend or foe was much more important than blending into the environment.

Sticking out like a sore thumb was the point - since troop formations had to be massive to concentrate the necessary firepower in an era before accurate or fast-firing firearms, there was often little interest in hiding those massive lines and columns. What was important, however, was figuring out who-was-who in the heat and smoke of battle - something made much easier if everyone is wearing easily identified colors!

As means of demonstration, in the US Civil War, when individual states were tasked with equipping and outfitting many units on both side of the war and had no set standards for what color or style to use, friendly fire was rampant because no one could tell who was on whose side. Hiding is less important than shooting the RIGHT people!

For this same reason, the American troops also largely wore bright colors in the American Revolutionary War - the main exceptions being militiamen, and skirmishers who were deployed ahead of the main force for harassing duties. The Brits themselves would adopt dark green for some of their specialized skirmishers shortly after the war.

The notion of obscuring troops instead of preventing friendly fire being a main purpose of the standard uniform is largely an invention of the late 19th century AD, when fast-firing single-shot rifles allowed even mediocre marksmen to pick off individual targets at 300+ yards. At that point, blurring one's figure, even just a little, becomes much more important. Into the 1st and 2nd World War, ever-faster and more accurate firearms and heavy weapons' support meant that combat increasingly became a game of hide-and-seek - he who is seen first, fucking dies. So camo is indispensable nowadays.

To add to the other good comments, at the time smokeless powder and radio didn't exist.
So as soon as an engagement started, the entire battlefield would be covered in thick smoke.
And without radio, commanders could only coordinate their troops' movements if they could see them.
So bright colors were essential to maintain troop cohesion and control.
Since muskets weren't very accurate and had a slow rate of fire, you relied on close formations shooting mass volleys at short range to achieve enough firepower. So you couldn't hide from your enemy during a battle no matter what you wear.

You've gotten answers pretty much covering all the reasons. I want to comment about Roger's Rangers, which is an example to show that military leaders in the 1700s were not ignorant of the value of camouflaged, stealthy units.

This was a British light infantry unit, starting as a single company and eventually expanded to a corps, that was first active in the French & Indian War. This unit was valued for its ability to scout, and perform unique missions like raids and ambushes deep in enemy territory.

The unit was active on the side of the Crown in the American Revolution, though some veterans joined the continental forces, meaning both sides used light infantry scouts.

Ranging units were a known and valued asset, however a military entirely made up of light infantry rangers wouldn't function. While these units could ambush and raid with the element of surprise on their side, in open field battles or static sieges they lost a lot of their punch. These units could deal damage and gather intelligence, but couldn't be relied on to hold ground against a conventional military assault.