。
She talked about her suffering with exaggeration.
Literal
She [topic-は] her own suffering [object-を] exaggerate [then-て] talked.
誇張して ('exaggerating') is the て-form of 誇張する functioning adverbially, modifying the manner of 話した ('spoke, talked'). 苦しみ ('suffering, pain, hardship') is the noun form of 苦しむ ('to suffer'), formed via the productive ~み suffix that turns adjectival verbs into experience-nouns (悲しむ → 悲しみ 'sadness,' 楽しむ → 楽しみ 'enjoyment'). The phrase carries a faintly disapproving tone — exaggerating one's troubles is generally not flattering.