Learn Swedish: Jag flyttar in i lägenheten imorgon. - I am moving into the apartment tomorrow.

The combination in i specifically shows motion from the outside to the inside of a space, translating exactly to into. If you just used i, it would mean in, which sounds like you are already inside the apartment while doing the moving. If you just used in, you would normally drop the destination, like saying just Jag flyttar in (I am moving in).

Why does the sentence use the present tense flyttar if the action happens imorgon (tomorrow)?

In Swedish, the present tense is very commonly used to talk about the future as long as there is a time word (like imorgon) that makes the timing clear. While you could say ska flytta (will move), using the present tense sounds very natural and shows that the plan is confirmed.

Could I use flyttar if I were just moving my physical body, like shifting on a couch?

No, flyttar is used specifically for changing where you live, or for changing the physical location of objects. If you want to talk about moving your body, you would use the verb rör (to move) or the reflexive phrase rör på mig (move myself).

How would this sentence change if I were moving out of the apartment instead?

You would swap the compound preposition to match the outward motion and say Jag flyttar ut ur lägenheten. The phrase ut ur (out of) is the exact opposite of in i (into) and is used when a person or object completely exits a three-dimensional space.