Beginning in the 1840s there was a great wave of German migration to the United States, which thoroughly transformed American society and culture. Among other things, German migrants made, sold, popularized and Americanized many foods, including a variety of sausages (“wieners”), sauerkraut, meat loaf, liverwurst, dill pickles, lager beer, seltzer water, pretzels, potato salad, Kaiser rolls, rye bread, and pumpernickel. The place where most of these items was sold was the “delicatessen”, a German word which essentially means “delicacies to eat”.

One of the items that was introduced toward the end of the 19th century was the hamburger, named after the northern port city of Hamburg. (One of the dishes associated with the city is the Rundstück warm, a bun sliced in half and filled with meat, which may have been the ancestor of the American hamburger.)

Both the Oxford English Dictionary and the Dictionary of American English identify the first use of the term "hamburger steak" in 1884, in the Boston Journal, although this referred to the steak itself rather than the combination of bread and meat. But there are also mentions of a dish called a “hamburger” in other periodicals as early as 1883 - The Caterer and Household Magazine listed a recipe for "Broiled Hamburg Steaks," adding that this "mixture of meat is also often spread upon slices of bread, with butter in which a spoonful of dry mustard has been mixed, and used as a sandwich." There is also a mention in the Chicago Tribune in 1896 of a “Hamburger sandwich”, sold in a “lunch car” on South Street:

A distinguished favorite, only five cents, is the Hamburger steak sandwich, the meat for which is kept ready in small patties and “cooked while you wait” on the gasoline range.

By 1908 (and possibly a little earlier) "hamburger steak" was shortened to "hamburger”; specifically as a reference to a bun filled with ground beef, the first mention of the term comes from 1912.

By the 1930s, Americans began playfully extending the term “hamburger” into other forms, leading to the shortened suffix of “burger” – possibly because they associated “ham” with the meat of the dish, and “burger” with what surrounded it.

Sources: Hans-Joachim Kann, “The ‘Burger’ Family,” Jahrbuch für Amerikastudien, 17 (1972): 213-215; http://aht.seriouseats.com/archives/2009/08/the-hamburger-was-not-invented-in-athens-texas.html; see also the discussion of the origins of hamburgers on Wikipedia (https://en.wikipedia.org/wiki/Hamburger#History)