How were these animals chosen over other similar species - buffalo, deer, horse, goats, sheep, ducks/turkey etc? Obviously some dishes highlight these other animals, however they aren't nearly as prevalent nor as factory farmed nearly as much as Chicken, Cow or Pig.
Not to discourage any further answers, but these older threads may be of interest to you:
Why did the chicken become a staple on tables around the world? by /u/xenofere
When and how did the domesticated cow spread throughout the world by /u/Valmyr5
[Edit: fixed hyperlinks]
I've raised livestock and have a degree in animal science from a college of agriculture:
I'm focusing on chickens in this comment because they are the easiest to address and are a relatively recent popular species.
If we look at the domestication process in general, we find some useful general information on the subject:
There is a list of soft requirements for animals to be domesticated:
Essentially, we need to be able to raise and eat them in reasonable time frames.
Everything about the animals we domesticate has something to do with ease of raising them; whether that is the actual managing of animals or the resources necessary (water/food).
There was an interesting relatively recent study on domesticating foxes that goes over it. Here is a scientific article on the subject and here is a BBC article. I believe there is a documentary or natural geographic type episode on the study that I am not finding at the moment.
Now to chickens specifically: FAO has some interesting stats on the amount of meat we (the world) eat and the relative change over the years. During their time frame of 1990-2012, we've had roughly a 33% growth in the world population while most livestock only increased by 13-20% with the exception of poultry. Chicken specifically has grown in popularity in the last couple decades with their headcount growing by ~104%. We're seen a relative decline in the number of livestock to the population except in poultry.
The world is getting smarter. We're able to get more food out of our animals due to better agricultural processes (better diets and better preventative medical care). Chickens take the cake though.
If we specifically focus on the US broiler chicken for a minute, we have some interesting results. In the last hundred years or so, we've seen a decrease in time to market by 58%, 150% growth in slaughter weight, ~2.5x improvement in the feed efficiency (an important agricultural stat measuring how much feed it takes to gain 1 unit of weight), and a 72% drop in mortality rate. Our chickens are growing larger, faster, on less resources, and are dying less (well, if you ignore the fact we slaughter almost all of them).
Chickens have a special trait compared to the other domesticated animals. A key metric in evolution and domestication is the generation rate; what is the time from an organism being "created" to the time it will create another organism. They breed often and grow quickly. This short generation time allowed for them to adapt quickly, changing their fat/protein composition to meet public demand, all while decreasing the cost. It is also an key trait for simply producing enough chickens to keep up with demand.
E. coli, a well known problematic bacteria, has a generation time of 15-20 minutes in a lab and 12-24 hours in the wild. That is why it is problematic: because it almost doubles in count every 12 hours. Chickens have a generation time of 23-28 weeks. This is very short in the animal kingdom. Thus, we can get ~2 generations of chickens in a year. Based on the fox domestication study, it takes somewhere around 10 generations to domesticate an animal, and likely somewhere around there to get a new breed. It only takes 5 years for chickens to create a new breed or complete a drastic change.
Chickens were not always the standard poultry; it is only in the last century or so that they became more popular. The originated somewhere in southeast Asia and spread across the world from there. They were already known and available, then the world kicked it up a notch with factory farming drastically increasing the amount of chickens we can produce.
There was also another important trend: there was a push against saturated fat, specifically in the US, starting sometime in the 1970s. NPR has a decent 5 minute conversation on it. Pork and beef is somewhere around ~5-6% saturated fat, skinless chicken breast is ~1% saturated fat. Chickens weren't always the low-fat animal they are now; that short generation time allowed them to adapt quickly to the change in market tastes.
We now have an animal that easily grows, easily reproduces, and is considered healthy.
Going back to the other animals:
Non-poultry animals generally require more feed and resources while reproducing less frequently and slower.
Ducks are difficult to raise requiring a larger water source.
Turkeys are native to the Americas, so most of the world didn't know about them until the last few hundred years. Turkeys also have a slower growth rate and produce drastically fewer offspring than chickens.
Currently, there is no other livestock that can as easily breed year round and provide as much meat with as little resources.
TLDR: Chickens only became the dominate, or staple, meat source in the last 100 years where feeding more people on less resources came into play. They rose in popularity due to better growth stats and adaptability.
Edit: rewording and clarifying a few things
Edit 2: I stumbled upon this article today. It does a better job discussing this from the true historical point of view (covering the last ~100 years) rather than the biological focus that I had. Tagging /u/caverunner17