No longer = not any more.
It indicates that something happened or was in a certain state in the past, but has now changed and cannot be described in the same, past, way.
It carries a meaning similar to used to, which describes something (an action or state) that doesn't happen any more (which has ceased) or is no longer true (I used to play football: action; I used to be fat: state).
Your original sentence can be paraphrased as:
I used to be confident, but I am not (confident) any more.
I was confident, but I am no longer (confident).