Learn Italian: Senza autostima, dubito di essere capace di affrontare il mondo. - Without self-esteem, I doubt I am capable of facing the world.

Because the subject of both actions is the same person ("I"). When using a verb of doubt like dubitare, you normally use che + the subjunctive if the subjects are different (like dubito che tu sia capace - I doubt that you are capable). But when the subject is the same, Italian replaces the subjunctive clause with di + the infinitive (essere).

Why is there a second di before affrontare?

The adjective capace (capable) requires the preposition di when it links to an infinitive verb. It works exactly like English here: capable of facing (capace di affrontare).

How is the word autostima constructed?

It is a compound word made from the prefix auto- (self) and the noun stima (esteem or respect). You will also see stima used on its own, usually with the verb avere, to mean having respect or high regard for someone.

What kind of "facing" does affrontare refer to here?

It means to face in the sense of dealing with, confronting, or tackling a challenge. It comes from the word fronte (forehead or front), so it carries the vivid meaning of meeting something head-on.