Just stared into a lemurs eyes for half a minute aparently (that felt like an hour) and it freaked me tf out. Since been staring into peoples eyes or animals eyes and its consistently weird and freaky no matter what. Also it feels like my conciousness is behind my eyes and maybe thats why they say eyes are the window to the soul.
Humans are social animals and were group hunters. Being able to judge emotions and intentions of another human just by looking into their eyes (silent communication) was a massive evolutionary advantage for us.
Facial expressions are a powerful form of communication. Faces (especially eyes) are so important to communication in the human brain that we see faces in places (pareidolia).
There is also a theory that we evolved to have the whites of our eyes visible so others can see where you're looking and react appropriately (especially in the case of danger).
I learned most of this in psychology classes years ago. There is also a Wikipedia article on the topic. I'm not an expert at all, but I find this stuff fascinating.