I look at a couple of things.

I usually read through something first without any sort of 'analytical strategy'. I pick up on things here and there that I like (from a stylistic or technical standpoint) but mainly I just read as a reader and see how it makes me feel.

Afterwards when I'm reflecting on what I just read or believe the work is worth taking another look at, there are a few very important factors I try to dig deeper into.

1: First one is voice. How is the author communicating? Is it through a first-person, stream of consciousness style? Third person omniscient? This is a pretty broad term that I also lump 'style' into. From there I ask what was the author doing specifically with his/her voice that was working for me? For example you might really like the authenticity of a first-person stream of consciousness novel you just read. Examining why voice is effective is both a matter of technical breakdown as well as appropriate slang usage, timing, and cadence. When I'm editing or critiquing someone else's work, voice is usually the first thing I look for. It's the most important, in my eyes at least.

2: Second one is character, and this is an obvious one. What about the characters made them interesting? What made me root for them? What made me hate or fear them? What made them believable? What were each character's motivations? From a writer's perspective I also look closely at verbal tics used by the author as well as inflection and tone each character uses frequently. Convincing dialogue gets me off.

3: Thirdly I examine the work thematically. This is a hugely important aspect to any affecting piece of narrative and one that stretches broadly over the course of a work, rather than something that can be picked up on immediately. For example in As I Lay Dying by William Faulkner (one of my personal favorites) a big theme in the story is identity. Who are we, really? How do we perceive others? What does it mean to have an identity? How easily can your identity be changed? If a work makes you ask questions and is reflected in the settings, characters, and voice, you know that the themes are strong. The more layered the themes are, the more complex a work usually is.

4: Plot is also something I look at, as (in my opinion) it can be the most difficult part to get just right. How do the events in the work move from one to another? How do they relate? Am I interested in the events happening, and am I invested in the outcome? Two sticking points for me as a writer are the beginning and the end. Does the beginning grab you? And does the ending satisfy?

This is a really broad topic and there's a lot to discuss here, but these are just a few of the elements I look for when dissecting a literary work with the intention of becoming a better writer. From there I usually try emulating writers that I admire on my own and see what about their style works for me and what doesn't. Then, after a bunch of reading, and millions of words written, you can start to find your own voice and try to put together a story with all of these lovely little elements.