Thanks for a nice question!
The participle qualifies a persona or an object in their relation to a certain action. Participles are mostly used in written language or academic context, and are less common in colloquial speech. They are used to compress texts.
There are 4 groups of participles:
Present active participle: читающий, who reads
Past active participle: читавший, who was reading
Present passive participle: читаемый, that was being read
Past passive participle: прочитанный, that was read.
Active participles qualify a person or an object performing an action by themselves:
Горящий огонь = Огонь, который горит
Burning fire = The fire that burns
Passive participles qualify a person or an object which are (were) the object of an action applied to them:
Подписанные директором документы = Документы, которые подписал директор
Documents, signed by the director = The documents which the director signed.
Participles are formed from verbs and are used as adjectives in the sentence.
In your example, "Не оставшийся во мне воздух" can be translated as "The air that didn't stay inside me".
"Оставшийся" is a past active participle formed from the verb "остаться".
Russian language teacher.