Learn Italian: Dove hai lasciato la giacca? L'ho lasciata al ristorante. - Where did you leave the jacket? I left it at the restaurant.

In the first sentence, the direct object (la giacca) comes after the verb, so the past participle keeps its default -o ending. In the second sentence, the direct object is replaced by the pronoun la (it), which comes before the verb. In Italian, when a direct object pronoun comes before a compound past tense, the past participle must change its ending to match the gender and number of that pronoun. Since la is feminine singular, lasciato becomes lasciata.

What exactly does the L' in L'ho stand for here?

It stands for la, meaning "it" in reference to la giacca (the jacket). When the singular pronouns lo (him/it) or la (her/it) come directly before a form of avere (to have) that starts with a vowel or an h, they drop their own vowel and attach to the verb with an apostrophe.

Since la gets shortened to L', how can a listener tell if the hidden pronoun is masculine or feminine?

By listening to the end of the past participle! Because the pronoun forces the verb to agree, the -a at the end of lasciata reveals that the hidden pronoun is the feminine la. If the speaker were talking about a masculine object, like il cappotto (the coat), the sentence would end in an "o" sound: L'ho lasciato.