Learn Russian: Я засмотрелся на эту красивую картину. - I got carried away looking at this beautiful painting.

The combination of за- and -ся added to a verb creates the meaning of getting completely absorbed or carried away by an action, often losing track of time. While смотреть just means to look, засмотреться means to get carried away looking or to stare in fascination.

Since verbs ending in -ся are usually intransitive, why does the sentence still point to an object (картину)?

You are right that -ся verbs cannot take a direct object in the bare accusative case. However, this verb takes a prepositional object. Because the base verb смотреть uses the preposition на + accusative to indicate the target of your gaze, засмотреться keeps that exact same structure: на эту красивую картину.

Is засмотрелся perfective or imperfective, and why does that matter for this sentence?

It is perfective. Verbs formed with this absorption за- ... -ся pattern are almost always perfective because they describe the result of getting sucked into an action. The speaker isn't describing the ongoing background process of looking, but the completed event of having lost track of time doing it.

Can I use this exact за- ... -ся pattern with other verbs of perception, like listening?