Chickens are very old in terms of domestication, and have had plenty of time to spread across the world.

The chicken genome was sequenced in 2004 and it was found that chickens were first domesticated in northeast India about 7,000 - 10,000 years ago, from the red jungle fowl. There are at least a couple hybridization events along the way in their domestication, the first with gray jungle fowl from south India (which gives chicken their yellowish skin) and probably another hybridization event with some other species of jungle fowl from India or southeast Asia.

The spread of chicken from India probably happened via the Indus Valley Civilization. IVC sites are rich in chicken bones, showing that these people bred chickens in great numbers. Two important events in domestication happened here. One was a change in a gene that regulates glucose metabolism, which allowed the birds to put on more bulk. And the other was in a gene that regulates the reproductive cycle, making it year long, so they lay eggs throughout the year instead of just in the breeding season.

The Indus Valley Civilization had extensive trade contacts with ancient Mesopotamia, and it's thought that chickens were thus introduced into Mesopotamia through this trade. There are cuneiform tablets from Mesopotamia dating to about 2,000 BC which refer to the "bird of Meluhha". Meluhha was their name for the Indus Valley people, so it's likely the bird in question was the chicken.

From Mesopotamia, chickens spread very quickly to Egypt, where they were raised for both meat and eggs. The Egyptians may have been the ones to first invent artificial incubation, where eggs are kept at a constant temp/humidity and rotated periodically to make them hatch, though it's possible that incubation was developed independently in other locations as well.

Chicken also spread to Europe from the middle east. The earliest evidence of domesticated chicken in Europe is from the Mediterranean, around 800 BC. The Romans loved chicken and bred them in huge numbers. But after the fall of Rome, the big farms disappeared and the art of chicken breeding was lost.

So chicken lost many of their desirable traits in medieval times. Medieval birds were smaller and scrawnier and laid fewer eggs. Birds like geese and partridge replaced them for the most part in Europe. And when Europeans traveled to the Americas, while they did take chicken along, they still remained a minor source of food. But turkeys were found in the new world, and they quickly became the more important poultry meat here, replacing European geese and ducks.

Of course, chicken retained their status as the premiere poultry in India, southeast Asia, China and also across the middle east. And then in the early 20th century, when techniques for factory farming were invented, there was a resurgence in the popularity of chicken in Europe as well. It turned out that chicken can be factory farmed very efficiently, and provide some of the cheapest meat available. Modern breeds of chicken have very high feed conversion ratios, approaching 2:1 (a kilo of meat for every 2 kilos of feed).

So although the spread of chicken across the old world is very ancient, and while it has been popular in various parts of the world at different times, the modern ubiquity of chicken is a relatively recent phenomenon dating from the advent of factory farming. And it owes to the fact that chicken is one of the cheapest sources of animal protein (other than insects) that can be produced on a large scale.