Food can be obtained from events, or found in Prize Boxes and Daycare Gift Boxes. They could formerly be purchased in the now-defunct Food Shop. They can, of course, be fed to horses of any stage, and will grant a number of levels specific to the food item that was used. Food can give anywhere from 1 to 149 levels at a time. Unlike on some other sites, there is no penalty for not feeding your horses, but feeding them can help them reach adulthood faster.
As of March 2019, Food items can be obtained as you navigate the site, having a random chance to appear each time you load a page.
Below is a list of food and how many levels each gives.