Over time horses will die of hunger if you dont feed them. You can prevent this by crafting horse fodder.
In order to craft one of the first fodder you need to hit level 14. Its called coarse dry fodder. Press “P” to open “Crafting & Recipes” and unlock “Tamed Horse”.
Go close to the horse and hold “E” to open the inventory. Drag the already built coarse dry fodder into the horse inventory. It will eat by itself.
Once your character hits level 13 you are able to craft common coarse fodder. Press “P” to open “Crafting & Recipes” and unlock “Tamed Horse”.
It feeds the horse just like the coarse dry fodder, BUT it restores health over time. Then dying is even harder. The higher your character level gets, the better fodder you unlock. The horse needs less food and restores more health faster.