I'm not a historian, but I am an avid amateur collector of antique cookbooks and I have a fascination with historical American cuisine. I can give you some history on these foods. The origins of popular foods in the U.S. can sometimes be obscure. Most of the time, a specific moment of invention can't be pinpointed, but instead fleeting references to them have to be chased through old cookbooks and personal diaries. People often just didn't consider food dishes to be important enough to record for posterity, except in a personal journal if it was really good or really weird. Dishes are often created out of necessity or imitation, by using what ingredients are easily available or trying to recreate something you've had somewhere else.
Most American fair foods came about in one of two ways:
Either they were a traditional festival food in various Western European cultures and were brought over here by immigrants, or
they were an 'exotic' or newly invented food introduced at a public spectacle like a World's Fair.
Some fair foods that came from European traditions and/or immigrants:
Candy apples have been served at carnivals, fairs and holidays for at least a hundred years. Caramel apples can often be seen as well, but the iconic type in American culture is the hard-coated, bright red apple on a stick. There is evidence that fruit coated in honey or sugary syrups has been a traditional harvest festival food in Western European cultures since ancient times. In France they are called pommes d'amour (apples of love), and the UK calls them toffee apples. The modern version, the bright red cinnamon flavored ones, were likely invented by a Newark, NJ candy maker during the 1908 Christmas season, were later sold on New Jersey boardwalks, and from there spread to the rest of the country.
Funnel cake was brought to America by German immigrants, specifically the Pennsylvania Dutch, who called it drechter kuche. The original versions were yeast risen, which took time and work to make. Variations of this dish are still served at festivals in Germany, I believe (any Germans around to corroborate this?) The innovation of the Pennsylvania Dutch was to use baking powder instead of yeast to make them, which cut down on the time and labor needed.
Kettle corn was also commonly eaten at gatherings in the 19th century, and is likely older than that. It was made in large cast iron pots for festive occasions, and has a traditional association with the Pennsylvania Dutch. It is also recorded as a popular food among cowboys in a number of diaries. It fell out of popularity and was obscure for a long time, but in the last few decades living history places have resurrected it, and it is appearing at fairs and festivals once more. If you've never had it, definitely try some, it is a revelation. It is sweet and salty and has a wonderful texture.
Corn dogs have an obscure history. They were probably created in Texas or the Midwest between the 1920's and 1940's. There are several competing claims for the invention, including German sausage-making immigrants in Texas and a vendor who sold them at the Minnesota state fair in 1941, but not a lot of evidence.
About the World's Fairs: One of the major purposes of these fairs, from a business perspective, was to show off technological advances and innovative products. The World's Fairs were major events for inventors and industrialists. This includes those who worked with food, and many fair-goers saw the fairs as a place to try strange things they normally wouldn't touch at home. Small samples of new or exotic foods tended to be a free or inexpensive treat, and so were a common thing for fair-goers to spend money on. Many of these foods weren't invented for the fairs, but popularized at the fairs, because businessmen wanted to create interest in or a market for their newly developed technology. The nature of World's Fairs has changed a bit in modern times, but the tradition of being inventive and coming up with unusual treats or strange food combinations for fair-goers to try has spread to state fairs. In the last few decades it has become popular to batter and deep fry all kinds of weird things like butter, Oreos, Twinkies and such, as well as serving things like Kool-aid pickles or doughnut burgers.
Cotton Candy is an example of a new technology introduced at a World's Fair. The machine for making it was invented in Tennessee in the 1890's, and it was originally marketed as 'fairy floss'. It was first sold at the Exposition Universelle (the 1900 Paris World's Fair), and later at the St. Louis World's Fair in 1904. Later inventors improved the machine and started calling it cotton candy. There were types of 'spun' sugar treats made in Europe going back long before this, but they tended to be expensive and labor intensive, so they weren't really available to most people. Like I said, what was introduced at the World's Fair was the technology to make it: Finely ground sugar was melted and spun in a large centrifuge with holes in it, where the sugar spun out into delicate threads and was collected onto a stick.
Peanut butter was was popularized at the 1904 St. Louis World's Fair. It had been available long before this, but was a delicacy mostly enjoyed by the wealthy; it was served as a health food by Dr. John Harvey Kellogg (yes, that dude who invented corn flakes to deter masturbation) at his fancy-pants Michigan Sanitarium. It was sold at the St. Louis World's Fair on crackers for a penny a sample and was hugely popular, resulting in a few companies marketing it widely in the years after.
Puffed wheat and rice were also shown off at the St. Louis World's Fair. Once an hour, workers would load grain into a specially made cannon, heating it until it exploded from the barrel over the heads of passerby. This technology was later used to make all sorts of puffed grain breakfast cereals.
A few others foods not invented for but popularized at World's Fairs include Dr. Pepper, Jell-o, waffle cones, Hershey chocolate bars, Cracker Jacks, Borden's Condensed Milk, Hires root beer, Pabst Blue Ribbon, Schlitz, Jack Daniels Whiskey, Log Cabin Syrup, bread made from Pillsbury flour, Heinz Ketchup, and French's yellow mustard.
Sources:
A lot of the primary sources for this information comes from random historical diaries and cookbooks I've read over the years and cannot remember exact details on. Here are a few actual published books, however, with good historical information that I used to write this up.
Beyond the Ice Cream Cone: The Whole Scoop on Food at the 1904 World's Fair by Pamela Vaccaro
Fair America: World's Fairs in the United States by Robert W. Rydell, John E. Findling, and Kimberly Pelle
Creamy and Crunchy: An Informal History of Peanut Butter, the All-American Food by Jon Krampner