Staring at some types of animals can be interpreted by the animal as a threat. The alpha male or female of some dogs and cats usually demand and get averted gazes from underlings. Unknown humans who stare down dogs or even cats either evoke a fight or a flight response from them. You can more often than not disarm animals by not giving them threatening or domineering eye contact but showing passive slow non advancing relaxed movements. Of course some animals don't wait around to communicate those actions, they either scram or they're in your face before you know it.

Not sure what to make of the chimp ban on staring though... If they are trying to integrate the anuimal into its own kind perhaps no contact at all with humans would be best.