Welcome to the boards Slavik!
I too have experienced difficulty looking people in the eyes, and I agree with Seleneccentric's view that it is a way of desiring not to be seen, being vulnerable.
I don't think you necessarily need to look people in the eyes, and certainly I would resist anyone who tells you that you need to, but what I will say is that same vulnerability and visibility that causes you discomfort or anxiety, that opens it up for the other person to see into you, is what makes you able to see into them. And that's enormously valuable to get unconsciously and quickly a whole lot of information about the other person.
It's really the same for vulnerability in general. If you are vulnerable, the way that people respond to that tells you more than probably anything ever could.
There is a kind of strength in vulnerability for that reason. You end up showing your cards, but at the same time (if you are paying attention) you see the other person's cards.
Barring some exploitative personality, if people are drawn to that, they are drawn to you for who you are, and if they reject that, then you can be sure that they are not someone you want around you.
As uncomfortable as it is, it gets easier the more you do it because you become more familiar with your own capacity to accurately judge people, and that in turn gives you extra self esteem.
Being vulnerable with people who haven't earned it can be dangerous though, which is likely the reason that it causes you discomfort: to protect you from people who would exploit you.