Learn Italian: A furia di crederci, ha incrinato ogni loro certezza. - By believing in it so much, he shook every certainty of theirs.

While credendo simply means "by believing", a furia di (or its exact synonym a forza di) emphasizes persistence or repetition. It means "by dint of", "by sheer force of", or "by doing [X] so much". As the current lesson notes, it must be followed by an infinitive (credere).

What is the ci doing at the end of crederci?

It translates to "in it". In Italian, "to believe in something" is credere a qualcosa. The pronoun ci here replaces a qualcosa. Because it follows an infinitive, it drops the final -e of credere and attaches directly to the end to form crederci.

The translation uses "shook" for ha incrinato. What does incrinare literally mean?

Literally, incrinare means "to crack" or "to fracture" (like a glass, a vase, or a bone). Figuratively, it is heavily used with abstract concepts like a certezza (certainty), an alibi, or a relationship, meaning to severely compromise, undermine, or "shake" it.

Why is there no definite article before the possessive in ogni loro certezza?