I don't understand how it works. Does it work on some specific occasions or it's always available. I tried to do it today and it says it can't find eligible games. I have nearly 400 games on Steam. However I had my profile on private. I set it to public, waited 2 hours, but still nothing.

Imagine it as a security guard at a club, restaurant etc., that only let's specific stuff enter when it wants, but otherwise keeps the door closed.
The "when" regarding connect being mostly during big sales as promotion and the "wich games" depends on agreements between gog and the publishers.

All your questions are already answered in the FAQ on the Connect page: https://www.gog.com/connect

I still don't understand. Can I do this process now or it's not available at this moment?

Games on GOG Connect are only put there at the request of the developers. Only a few titles appear on there and usually only during the two major summer / winter sales. It's never been a thing to get a free GOG version of your whole Steam catalogue. So right now, it's unavailable (which is normal for +48 weeks of the year).

Specific games are only offered at specific times in short intervals. Right now the only game available is Legends of Amberland: the Forgotten Crown. If you don't own that one on Steam, then there's nothing to claim.

Joined GOG in 2016, has only two games in his account, of which the only one that's visible is Gwent, which is free.

When was GOG connect introduced? Oh, in 2016...yeah - that fits.

As it says right on the GOG Connect page: " The eligible games are limited-time offers made possible by participating developers and publishers".

GOG Connect is an occasional promotion not a permanent feature. If there are no games on the Connect page, it's because there is no Connect promotion active at the moment.

Oh, I see. Thank you.

What is gog connect :O?

It’s a method of transferring rent wares from steam over to full user control.

I see , where can i see the list of games I could transfer?

here: https://www.gog.com/connect

Currently no games available

Thx
Ah thats why I havent seen anything and thought the list is elsewhere :)