I like the scientific approach, that primates are evolutionary capable to identify when some other is looking at you, but I would like to believe, that it means a lot more than that, let me explain.
First I like more the phrase of “attention” than “looking”, because if you pay attention on somebody means, that you care about him, you fill your mind with the fact that you observing.
You can even discover someone’s attention even if he or she is not in your field of view (e.g. behind you). You just simply feel that feeling “someone is looking at me” and turn your head toward to the person. It is completely unconscious.
Recent scientific discovery in field of quantum physics identified that elementary particles (electrons) would change their behavior (whether they are in wave or particle function) just by observing them. It suggests, that observing makes difference in the nature of the small particles.
After a talk with one of your friend you will feel completely different if he or she look into your eyes, you will feel filled up with energy.
I taught students at university some years ago, in the field of ERP systems, and I noticed that it was quite different, if I looked into the students eyes during the speech, and yes it was exhausting to keep the eye contact during the lecture, but the information flow was much stronger, and I attracted much more attention on the class.
There was an interesting experiment performed by Professor Arthur Aron (State University of New York at Stonybrook). He put strangers of the opposite sex together for 90 minutes and had them discuss intimate details about themselves. He then had them stare into each other’s eyes for four minutes without talking. In result many of the subjects felt a deep attraction for their partner after the experiment, and two even ended up getting married six months later.
To tell the truth, I really like the rational explanations, but my sense of justice, does not let me verify it is only matter of behavior and evolution. Something happens, when we look in each other’s eyes, this is the situation where words are worthless, where there is a connection between us, where our everyday ego does not work anymore, and where we could catch a glimpse of each other’s inner world. It happens from time to time.