Several advantages made muskets very attractive.

First and foremost, both bows and muskets, and ranged weapons of any kind for that matter, need to be fired in volley to be effective. Without coordination and concentration, skirmishers will be nothing more than a nuisance. In order to obtain enough skirmishers, then, you need a weapon that is simple and easy to use. A longbow requires immense strength and concentration to pull back and aim, and a lot of skill to hit a target. A musketeer, meanwhile, can be trained in a fraction of the time, because he is using a weapon that requires almost no skill to reload and fire. Thus, the large amounts of skirmishers needed to have an effect on the enemy can be raised in a fraction of the time with a fraction of the cost.

This was the main reason (if you want more, research the Battle of Nagashino and Oda Nobunaga's use of arquebusiers against a cavalry charge), but there are others. Muskets make a lot of noise and smoke that can scare inexperienced troops and horses. Their trajectory wasn't as affected by wind as a longbow's, and musket balls were cheaper and easier to make on the march than arrows.

In addition, much about the longbow is exaggerated. It wasn't invincible, and the range listed here at 200 yards would have done nothing against good armor. The longbow's victory at Agincourt was at almost point blank range; the same effect could be achieved with a musket, which had far higher stopping power and penetration, at a longer range. Also, it is worth noting that as the range and price of the musket got better, its usage in Europe skyrocketed while the longbow's performance remained pretty much the same.

True, the longbow was an amazing weapon. New and improved compound bows can rival rifles in deer hunting competitions today. But it just couldn't compete with chemistry, and gunpowder won the day.

TL;DR: Longbowmen were too hard to train and too expensive.

I'm afraid the other poster isn't entirely correct.

The idea that firearms replaced older missile weapons because they were cheaper and easier to learn might have been the case for some regions/periods, but it doesn't seem to hold true for Tudor England. Laws requiring archery practice remained in place throughout the 16th century and government regulations ensured that even a good quality yew longbow remained much less expensive than a matchlock firearm.

In general the English military was relatively slow to modernize. It wasn't until the 1530s that England started really using guns, and by the time the 1544 invasion of France was launched the bulk of the English army was still made up of traditional county levies armed with bows and billhooks, forcing King Henry VIII to hire expensive foreign mercenaries armed with pikes and arquebuses and to offer increased wages to English soldiers who were willing to take up the new weapons. Accounts from the fighting around Boulogne seem to indicate that the English archers were now being badly "outgunned" by French infantry. According to one Welsh captain, "I never saw Welshmen or Englishmen so bad hearted or so unventuresome as I saw at this time. Not a single one of them would dare to go near where the handguns were shooting at us." Humfrey Barwick, an Englishman who had spent his youth learning archery before being given an arquebus when he joined the army, claimed that "I did never see or hear, of any thing by them don with their long bowes, to any great effect. But many have I seene lye dead in divers skirmishes and incounters [from harquebus and pistol bullets] . . ." From the French side veteran Blaise de Monluc noted that there was a fair amount of respect for the English and their bravery when the conflict began, but after skirmishing with them for a while, he concluded that there was little to fear since the english carried "arms of little reach" compared to the French harquebusiers and they could be made to turn their backs "with as great facility as any Nation that ever I saw". After the war ended the proportion of firearms to bows in the english army would continue to rise until the longbow was officially abandoned in 1595.

During the last couple decades of the of the 16th century, the subject of longbow vs gun became a pretty hot topic among English military writers and numerous books and pamphlets were published which discuss the subject. Giving us a pretty good idea about the reasons given for the longbow's replacement. Some were still opposed to giving up what they saw as the traditional English weapon, pointing out how instramental it was to battles like Crecy and Agincourt. Others, including well known veterans like Sir Roger Williams, Barnabe Rich, and William Garrard, considered the longbow fairly useless except perhaps as a militia weapon or something to be used when not enough firearms were available. The crux of the debate tended to involve on arguing the technical merits of each weapon, with the pro gun side pointing out that a musket could shoot a bullet many times farther than the best archer, muskets could be used more effectively from behind cover, a musket was aimed "by rule" rather than "by guess", bullets caused greater wounds, and while a bow could shoot more rapidly a musket could be loaded with multiple bullets at once. According to Lois Schwoerer in Gun Culture in Early Modern England the works supporting guns were overall "more numerous and more persuasive."

As for one last point related to training. One claim that was often brought by the pro-gun side was the fact that archery practice was slowly decaying and there seemed to be fewer good archers available, but it was typically brought up alongside other arguments about the technical superiority of firearms. Barwick's argument was that after just four months of practice with his arquebus he could shoot better than he could after years of practice with the bow and was on par with the best archers in england. Notably absent, on the other hand, were many treatises which consider the longbow the superior weapon, but argue that it should be abandoned anyways due to the lack of trained men. In fact, archery proponent Sir John Smythe, who did consider the longbow superior, completely discounted the idea that there were no good archers any more, pointing out that in any period some men are naturally born stronger. It's worth noting that of the skeletons found aboard the 1545 wreck of the Mary Rose, only 1/5th showed changes consistent with longbow practice from a very young age, yet all the bows had a 100+ lbs draw weight, which should have been more than enough for the late 16th century when relatively little armor was worn. Steven Gunn in "Archery Practice in Early Tudor England" concludes from records of archery accidents that there was some decline, but it seems to have been a fairly slow process over the course of the 16th century.

Keep in mind, that the debate at this point was not over whether muskets should be used, but whether Longbowmen should continue to be employed at all. If the problem was, as Roger Williams claimed, that only a fraction of the archers could make "good strong shoots", then Smythe would have had it that that fraction continued to be employed as archers alongside the musketeers. Yet it was extremists that considered archers completely useless who ultimately proved the most persuasive. And in 1595 the Privy Council decided that all remaining longbowmen should be reclassified as "unarmed men".

Sources:

"The commentaries of Messire Blaize de Montluc, mareschal of France" by Blaize de Montluc (1500-1577)

"A breefe discourse, concerning the force and effect of all manuall weapons of fire and the disability of the long bowe or archery" by Humfrey Barwick (1592)

"A briefe discourse of vvarre" by Sir Roger Williams (1590)

"The arte of vvarre Beeing the onely rare booke of myllitarie profession" by William Garrard (1591)

"Certain discourses" by Sir John Smythe (1591)

"A right exelent and pleasaunt dialogue, betwene Mercury and an English souldier" by Barnabe Rich (1574)

"Longbow and Hackbutt: Weapons Technology and Technology Transfer in Early Modern England" by Gervase Phillips

"Archery Practice in Early Tudor England" by Steven Gunn

Gun Culture in Early Modern England by Lois G. Schwoerer

Elizabethan Military Science by Henry J. Webb

Weapons and Warfare in Renaissance Europe by Bert S. Hall

Bow vs Gun introduction by E.G. Heath

Certain Discourses Military introduction by J. R. Hale