The repetition of “still” here takes on two meanings of the word. As an adjective, “still” can mean not moving or making a sound. As an adverb, it can also refer to time spent doing an activity, even now. In the repetition, the speaker says that he lies still in order to continue to hear her breath. In a sense, his only way to combat the progression of time is to make his body as motionless as possible. Notice the subtle irony that underlies this statement: the speaker must become motionless to make the moment last longer; he must mimic death in order to gain the feeling of everlasting life.
