Learn Russian: Открыв новую книгу, я зачитался допоздна. - Having opened a new book, I got carried away reading until late.

This verb illustrates your current lesson! The prefix за- combined with the reflexive postfix -ся creates a verb meaning you did an action to excess, usually losing track of time. Added to the root of читать (to read), it becomes зачитаться (to become completely absorbed in reading).

Why use the form Открыв instead of a regular verb phrase like Когда я открыл?

Открыв is a perfective verbal adverb meaning "having opened." While you could use когда я открыл (when I opened), using открыв is a much more elegant and common way in written or higher-level spoken Russian to show that the first action was fully completed before the main action (зачитался) began.

Why does the word допоздна end in -а?

Допоздна is an adverb meaning "until late." It is formed from the prefix до- (until) and the root of the adjective поздний (late). When forming adverbs with the prefix до-, Russian uses the suffix -а, which mirrors the genitive case ending that the preposition до normally requires.

Would it make sense to use the imperfective зачитывался here instead?