Often when I'm in a public place, I get the intuition that someone is looking at me even when I don't turn or look in that direction or move my eyes. I just have a strong feeling in my heart that someone is gazing at me. I immediately want to turn into that direction and many a times when I do , I do find a person gazing. Sometimes , I don't. Is it me overthinking or some kind of extra vigilant behavior?
There's a history of being amazed at similar ability that has led to some extra-scientific theories about it (see for example "psychic staring effect" on Wikipedia).
A better explanation is that gaze is very important to survival in the environment, and even more so for social species like humans. Even infants show sophisticated gaze detection behaviors (see Farroni et al 2002). Humans and other animals that are looking at you might be interested in eating you, fighting with you, or mating with you (see Emery, 2000). It's important to detect gaze from the entire visual field (including the edges of vision in the periphery) because you don't necessarily know where threats or opportunities may come from. You don't need to be seeking out gaze to notice it, just like you don't have to look for a brightly flashing light or an object moving towards your face.
Gaze detection is pretty ubiquitous among animal species with sophisticated visual systems. A previous asker at Biology.SE asked this question: https://biology.stackexchange.com/questions/95366/ after noticing birds who were quite tolerant of others in their environment until they were looked at, at which point they flew off.
Emery, N. J. (2000). The eyes have it: the neuroethology, function and evolution of social gaze. Neuroscience & biobehavioral reviews, 24(6), 581-604.
Farroni, T., Csibra, G., Simion, F., & Johnson, M. H. (2002). Eye contact detection in humans from birth. Proceedings of the National academy of sciences, 99(14), 9602-9605.