Why do so many people worry about their competences to perform the tasks they get?

You are hired to do the kind of work that gets assigned to you and not to worry if you are qualified to do it. Unless you are in a shitty* company this is someone else’s job to worry. I see people doing this to themselves and frequently have to let them show the value of their work. Many times before they understand what I see in their contributions.

Stress is fine, it will help you get further. But only to a certain point. If you don’t have faith in your capabilities, have faith in the management team...

* if you are in a shitty company, you should adjust your priorities. Do not worry too much, learn as much as you can and seek other options.