Just like you, your pup has tear ducts that produce real tears. When his tear ducts doesn't properly drain his tears, the result is reddish-brown staining under his eyes. Other symptoms of tear duct issues include dampness beneath his eyes and pawing or rubbing the same area.
Epiphora is a symptom of a condition whereby the eye produces too many tears. It's worth having a vet check to see if a reason exists for his extra tears. Allergies, conjunctivitis, abnormal eyelashes, eye infections or injuries, glaucoma, corneal ulcers and anatomical abnormalities are some causes of improper tear drainage, leading to staining.
Whether your pup's condition is chronic or the result of something temporarily irritating his eyes, you have several options for cleaning up his face. Your vet can recommend commercial products, including wipes. Warm water or an eyewash made of chamomile, calendula or goldenseal will work to clean up tear stains. Simply moisten a paper towel with the liquid, and gently wipe the stains from the affected area around his eye. Avoid bleach and hydrogen peroxide as eyewash ingredients, as they can be harmful to your pup's eyes.