To put your case well during a job interview you need to tell the panel what you have to offer, expressed in the most beneficial light. This is what self-promotion is all about.
You may be thinking it is something else. I’ve found people confuse self-promotion with:
Saying they are better than others
Telling a selection panel what you have to offer is not the same as saying you are better than others. There is no comparison involved. You are saying what you have to offer. It is up to the panel to decide if what you offer is more interesting/attractive than what someone else offers.
Being an obnoxious skite
There is a vast difference between confidently saying what you have to offer and bragging in an exaggerated, know-all kind of way. For a person who is reluctant to say much at all, there is very little risk that you will transform into one of these people. What you do need to be willing to do is move to a point where you can state what you have to offer in sufficient detail so that the panel can assess your claims.
Unnecessarily telling the panel what they already know
If your boss is on the selection panel it may seem like you are telling people what they already know about you. The problem with this is that even if they do know what you do, you still have to tell them during an interview. While your boss may know, in general, what you do, they are unlikely to know in the detail you imagine. No one knows what you do as well as you do.
So when you attend an interview you need to go prepared to say what you have to offer in a confident manner, in enough detail, at the right level of detail, with a link to the job you are applying for.