まま following a verb ta-form means “while doing something” or “in a certain state”.
立った is the ta-form of a verb 立つ meaning “to stand up” so 立ったまま means “while standing up”.
There is a similar word to まま which is ながら. It’s also translated as “while doing something“ but they are not usually interchangeable.
ながら following the stem of a verb masu-form is used when two actions take place at the same time, whereas まま is used for an ongoing state that exists while the main action is taking place. So 立ったまま寝てる shows that she is in the state of standing up while doing the main action.
Lastly, 寝てる is more casual than 寝ている which consists of the te-form of a verb 寝る meaning “to sleep” and a subsidiary verb いる meaning “to stay still”. A verb te-form followed by いる describes a continuing state resulting from the earlier action. So you fall asleep (寝る) and the state of sleeping is continuing (寝ている) at the time of statement.