No one really knows. During ancient times, Greek and Roman goddesses were sometimes shown sitting aside on horseback.
You may have seen pictures of the Virgin Mary sitting sideways on a donkey. This is probably correct. If Mary rode a donkey, she would probably had sat on a saddlepad. This was just a cloth or blanket. It didn't have the wooden foundation, called a tree, that saddles have today.
The first true "sidesaddles" with a tree developed in the the Middle Ages. You've probably seem movies and pictures of knights. Knights depended on their large, secure saddles to help them stay on their horses during battle.
Knight's saddles made very secure sidesaddles. With the addition of a wooden footrest and sometimes even a backrest, a woman could feel safe sitting sideways on a horse.
Medieval women didn't ride sidesaddles all the time, though. It appears that they sometimes rode astride (like men). For ceremonies, though, wealthy noblewomen rode sidesaddle so that their dresses could be seen!
On this type of saddle, a woman could face forward while sitting sidesaddle. These saddles often still had a wooden footrest rather than a stirrup.
Left: Alix of Brabant rides sidesaddle while hunting. This picture is from a French royal seal of office.