Learn Swedish: Lägg dig på ryggen och andas ut. - Lie down on your back and breathe out.

The Swedish verb for "to lie down" is the reflexive verb att lägga sig (literally "to lay oneself"). When you give a command to someone, the implied subject is "you", so the reflexive pronoun changes to match it: dig (yourself). Without dig, the imperative lägg just means to put or lay an object down.

Could I use the verb "att ligga" and say "ligg" or "ligg dig" instead?

No, because att ligga describes the state of already resting horizontally, while att lägga sig describes the physical movement of getting into that position. You would say ligg stilla (lie still) to someone who is already on the floor, but to tell them to get down there, you must use lägg dig.

Why does "andas" end in an "s" if it is an imperative command?

The verb att andas (to breathe) is an s-verb. These are special verbs that always end in an "s" in every single form, including the infinitive, present tense, and the imperative. There is no form of this word without the "s".

Why does the Swedish sentence say "ryggen" (the back) instead of "din rygg" (your back)?

In Swedish, you usually use the definite form of a body part, like ryggen, when it is already obvious whose body part it is. Because you are telling the person to lay themselves (dig) down, it is perfectly clear from the context that they will be lying on their own back.