Learn Spanish (Spain): De repente, a ella le dio por coleccionar sellos.

The expression darle por acts impersonally here, meaning the grammatical subject is an unspoken "it" (the sudden whim or caprice). That "whim" hits the person, which makes the person the indirect object (le). Literally, it translates closely to "it gave her the whim to...".

How does darle por differ from just using empezar a (to start)?

While both mean starting an action, darle por specifically implies that the new habit is a sudden whim, an unexpected phase, or even a bit eccentric. Empezar a is completely neutral. The phrase De repente (suddenly) perfectly sets up this impulsive mood.

Why do we have both a ella and le right next to each other?

This is a common Spanish feature called indirect object doubling. The pronoun le is structurally required by the phrase le dio por. Because le can mean him, her, or you (formal), a ella is added to clarify exactly who had the sudden urge, while also adding a natural emphasis.

Why is the verb coleccionar in the infinitive when the English translation uses "collecting"?

In English, we often use the "-ing" form after prepositions. However, in Spanish, prepositions like por must always be followed by the infinitive form of a verb. You cannot use the Spanish gerund (like coleccionando) here.