Ouija boards, as their old-time-y design suggests, first became popular in the United States in the late 1800s. The nation was still recovering, economically and psychologically, from the Civil War, and the spiritualist movement—the belief that one could commune with the spirits of those who had died—had grown in a nation where so many people had lost loved ones.
In 1886, papers around the country published an article from the newly formed Associated Press, describing one of the products of that movement in Ohio: "talking boards" that claimed to help the spirits to send messages to the living. In 1890, the businessman Charles Kennard formed the Kennard Novelty Co. to make and market talking boards. The company would be joined in short order by William Fuld, who—largely because of his savvy marketing of the device—would become known as "the father of the Ouija."
Fuld was intentionally mysterious about whether the board worked, how it worked, and whether it was just a game or a more serious spiritual tool. This marketing strategy was extremely successful, and everyone had to have an Ouija board. People played at home with their families, including their kids, which may seem strange now.
In the late 1800s continuing all the way into the 1960s, the Ouija board was considered good, clean, family fun.
The other thing the Ouija board had going for it was that it could be, in its own special way, romantic. Flirty, even! At a time when there were strict regulations about how (and where, and when) courting couples could interact, the Ouija boards provided an excuse for physical proximity that was also, in its own special way, chaste. The boards were initially designed to be placed in one's lap. And even when the board was placed on a table, participants would generally sit knee-to-knee around it. The planchette—the vaguely heart-shaped device used for pointing out letters and numbers on the surface of the board—required participants to touch their fingers as they navigated it across the board. The Ouijas allowed for a very Victorian version of flirting.