Originally, it was simply a bowl of chili. If you see the item today (which is rare, since the term is almost obsolete), it's usually an open-face chili burger.
From 1913-1958, there used to be a 24-hour chili parlor called "Ptomaine Tommy's" in Los Angeles. It was the largest and best known chili parlor in town. Ptomaine Tommy served straight chili and a Southwestern variation, a hamburger smothered with chili. He had two ladles, a large and a small. When a customer ordered straight chili, he got out the large ladle. When he wanted the other, he usually said "Hamburger size." So Ptomaine Tommy put up one sign that read HAMBURGER SIZE 15¢, and another that read CHILI SIZE 20¢. Other chili joints followed suit and before long chili was known throughout Los Angeles as "size". They'd say, "Just gimme a bowl of size."
Since Tommy created perhaps the very first chili burger, other restaurants began copying it. They thought the new item was called a CHILI SIZE. So, that's why, today, you might find a chili size on a diner menu that consists of a burger patty topped with chili, cheese, and onions.