"Anna Karenina" is a Russian novel, translated to English. "Expanding his chest" might be the best translation of a Russian idiom, but it's not idiomatic in English.
We might say someone "puffed out his chest" to indicate confidence or braggadocio, but in this case the translator should have substituted a more appropriate English idiom like "he gathered himself together" or "he took a few deep breaths", to show he's trying to "regain his composure".
In any case, "expanding his chest" is a participle phrase that acts as an adjective (not a verb) which modifies the pronoun "he". Other examples:
Gathering his strength, he pushed the heavy rock out of the way.
Drying his eyes, he resolved never to fall in love again.
Searching his pockets, he found a few dollars for bus fare.