。
Poverty drove him to stealing.
Literal
Poverty [blame-のせいで] he [topic-は] theft [object-を] doing [came-to-ようになった].
のせいで assigns blame — 'because of (and it's the fault of).' More judgmental than neutral のため. 盗みを働く ('to commit theft') is a set phrase where 働く means 'to perpetrate' rather than its usual 'to work.' ~ようになった marks a gradual change of state — he 'came to' commit theft over time.