The riddle
Why Did the Chicken Cross the Road?
A joke that refuses to be one
The listener expects a pun or a twist. What arrives is the most obvious reply possible: the chicken crossed the road to get to the other side. That is true of anyone who has ever crossed a road, which is exactly the point. A joke that gets its laugh from a punchline that is not funny is called an anti-joke, or anti-humor, and Wikipedia's article on this riddle names it as a standard example.
The 1847 printing
Barry Popik, a researcher of phrase and joke history, traces the earliest known printing to The Knickerbocker, a New York monthly magazine, in March 1847, on page 283. It asks, "Why does a chicken cross the street?" and answers, "Because it wants to get on the other side!" The passage around it, as Wikipedia quotes it, is about "quips and quillets" that seem actual conundrums but are none.
In a 1924 collection
J. Gilchrist Lawson's 1924 collection has a version with a hen: "Why does a hen cross the road?" The printed answer is "To get on the other side." Popik also lists a Wisconsin newspaper, the Appleton Crescent, printing the question with a road in February 1869.
Why the answer always fits
Every road has two sides, and a crosser starts on one and finishes on the other. Merriam-Webster defines cross as going from one side of something to the other, so the answer repeats the question's own verb. No other reason is needed, and that is what makes the riddle a perfect fit for any animal or person.
Follow-ups worth knowing
A playground version, which Wikipedia mentions, has the chicken cross a playground to reach the other slide. For a bird that does it with a different reason, see why did the turkey cross the road.