Schraf likes to call it your conscience.
I disagree. I can tell the difference between my conscience, and God most of the time. Usually God is your first impression, then your conscience takes over.
When your doing stuff, and your mind is occupied by the task at hand, your mind is clear, and God can come through. It's when you stop the task, and then say wtf? was that feeling I had, and then justify it away.
I guess you could start asking yourself, or start praying to God, to reveal to you why you feel this way.
It could be something from your childhood, or your upbringing that makes you feel like that. If your asking God, and He reveals that to you, it may possibly be something you need to be released from. I pray a prayer of "good enough" over you right now.
Or you could relate it to God, in the sense that, the bible teaches us that we were created to worship God. We do this in spirit and truth. We are also to do things to God to the best of our abilities.
This feeling you have of being watched may be a gift from God, in that you strive for excellence. Not everyone is like that. God created you for that, and he is always looking over your shoulder, to see how your doing with it. Maybe one day you'll look back and say, how's that? That's how we give glory to God.
It would seem we are born with a sense of "something more". People who don't believe in God, will say it's just because there is more. Other's will think it's God. To me, that is our free will, deciding if it's God or not.
So it's not a fact that I can tell you about, it's your choice.
What leads me to this? Why God of course, He's the one who calls us, and looks over our shoulders.