It happened gradually over hundreds of years. The beginnings of gunpowder weapons are in China in the 10th century, originally more like a flamethrower device (source 1). The Medieval era saw development of cannons for sieges, although hand cannons and early guns did exist. The Mongols or travel along the silk road brought gunpowder to Europe and the Middle east. Early gunpowder weapons were unreliable and dangerous. The Arquebus and wheel lock muskets would be the first firearms that bears resemblance to modern firearms. It used a match lock or a wheel lock device to ignite a pan of powder that would then light the powder inside the barrel. But even this would not be the end of melee combat. The armies of 15th, 16th, 17th, and 18th centuries would still depend on melee troops to take the field of battle. By the 18th century however, bayonets were affixed to muskets to make them into short pikes. Taking the field of battle would still need melee to route enemy forces, however this was done with bayonets. Calvary would use single shot pistols and sabers, often being used to flank enemy infantry and artillery. The 19th century would still see this happening well into the middle half of the century. The Napoleonic wars saw the high of this style of combat, where muskets where used until a general sought to take the field . Notably the charge of the Light Brigade, in the crimean war, although ill fated and advised was an example of the use of melee calvary. The American Civil war would also see Melee Calvary and infantry charges using melee combat to win the field as well. However, the beginnings of widespread rifle usage meant that melee to take the field ended up in huge casualties. The late 19th would see the development of the first machine guns, such as the maxim gun, that would help to make melee combat as the way to win the field of battle obsolete. By the time of the first world war, Melee combat was not a feasible option as machine guns could mow down oncoming attackers. That did not stop the generals from attempting to use napoleonic tactics at first though. The development of combined arms tactics by the end of the war meant that melee combat would be far more limited in military endeavors. With tanks, submachine guns, and self loading rifles, it simply was no longer a viable tactic to employ melee combat as a means to route the enemy. It can still occur, in close combat situations it can be more viable, but it is far rare as assault rifle, submachine guns, and automatic pistols have proliferated. It should also be noted that the cost of firearms and gunpowder fell as the industrial revolution took hold. The cost of training someone with a musket was far lower than training someone for melee combat. The advent of cheap guns meant that more of them would be used in combat. This started in the 18th century and really picked up in the 19th century. Additionally standardization and new machining process enabled high quality weapons to become affordable and easily repairable. This helped end melee combat as these new weapons had longer ranges and were more accurate. Hopefully that answers your question
Edit: Realized i forgot to put the word obsolete in regards to 19th century developments
Sources
fyi you'll find lots more on this transition in the FAQ