Horses were actually replaced rather slowly. Local freight was still hauled mostly by horse until the 1920s. Automobiles were the province of the rich until they were mass produced, and even after the advent of Fordism many people kept their horses. And so, it was instead a matter of selling a few less horses each year in a slowly dwindling market rather than an issue of ridding society of millions of horses at once.

Fun fact, late 19th-century experts worried a great deal about the environmental consequences of horses and thought that by the next century cities would literally be covered in a layer of manure 9 feet deep. Removing dead horses from massive urban centers like New York was no picnic, either, and drove early civic hygiene and sanitation efforts. New York was faced with over 20,000 dead horses a year by the 1880s, part of a hideous cocktail of putrid smells.

I'm currently pouring through newspapers in the Midwest of the U.S. in the 1890s and I can assure you that horses were still very much in demand. Cars were extremely rare and railways only went to specific stations. Getting around a city or traveling between rural towns was all done exclusively by horse or on foot.

As the other poster (whose username is unpronounceable!)- it was a very gradual change and there was never- in the US at least- sudden need to get rid of a large number of horses.

Source: Cincinnati Enquirer, Kentucky Post, Louisville Courier-Journal, and New York Times of the 1890's.

Horses actually remained critical to military logistics well into the 20th century. During World War 2, as much as 80% of the German Army's logistics train was horsedrawn.

Side question: what did armies do with their horses? Because this movie is sad.