Horn is collected by killing and harvesting Deers or as a passive income from a Deer Pen. Its a essential material only used to craft the Scythian Bow and Standard Horn Bow.
The slower method to get it is to kill and harvest Deer with a Stone Hunting Knife. They are very fast and run away. Craft a Bow to shoot them from far or Tame a Horse to attack them while riding. The probability of obtaining Horn from harvesting are very low, but the chance is there.
The much more effective way to get Horn is by building a Deer Pen. Catch and breed Deers to get a permanent passive income of Horn and other resources if you feed them. Once you hit level 31 you can unlock it. Press “P” to open “Crafting & Recipes” and unlock “Medium Cage”.
Check out this guide to learn more about Pens and how to tame and breed Deers.
The more animals are in the Pen, the more ressources you get. You can build Pens for every animal in the game except of horses, elephants and rhinos.