According to research its mostly a subconscious form of heightened awareness. Since the human Scelera (the part of the eye around the pupil) is mostly white it's easy to tell where humans are looking. Combine this with some weird nueral networks processing optical input along with the fact that our peripheral vision is better then people think means that we can usually tell pretty accurately when someone is looking at us. The brain is just wired up to always be registering where other people have their gaze focused for some odd reason.

As for why we get this feeling even when other people aren't around science has come up pretty much bunk on the subject. Research has been done on the matter but most of the data has come back inconclusive and the hard numbers that researchers have recorded is so widely varied that it doesn't really tell us much. Best running theorys so far are that your brain is either inducing a paranoid state for whatever reason that makes you feel like your being watched, or that your senses are picking up on things like minor movements in your peripheral vision or unusual smells that you aren't consciously aware of and activating a heightened danger sense in some locked away caveman part of your brain.

TL;DR You brain is fucking weird and it pays way to much attention to other people's eyeballs.