I myself have experienced it before. But the answer is rather confusing to know.
Sometimes I might misinterpret wrongly. But I thought to myself and I know I really do feel that it is what I feel.
1. People look you deep in the eye because he/she admires you
2. People look you deep in the eye because he/she wants you but could not
3. People look you deep in the eye wondering if he/she understands what he/she is talking about
4. People look you deep in the eye because he/she is so angry but could not let it out
5. People look you deep in the eye because he/she is hurt and wants sympathy
Which is it? Does anyone have the real answer?