You get the leading feature when your horse reaches level 3. The feature needs to be unlocked for each individual horses.
Once the lead feature is unlocked, all you have to do is hop off your horse, and then either stand by its head and click the lead button that will appear on your screen down by the horseshoe menu. Or you can click on the horse's head to access the new horse care menu. There you will see an option to lead your horse.
Keep in mind that you can only walk with your horse when you are leading it. Something else that is good to keep in mind, is that certain halters that don't have a lead rope or reins attached to them won't show a lead rope when you are leading the horse.