Having a prominent or well-developed chest; or showing arrogant self-confidence and pride in a bold way.
From 'chest' (the front part of the body, from Old English 'cest') with the adjective-forming suffix '-y.' Originally literal (having a big chest), it evolved to mean confident or arrogant.
The journey from 'chesty' meaning physically broad-chested to meaning full of yourself is a perfect example of how physical traits become personality traits in slang—kind of how 'thick-skinned' moved from literal skin to emotional resilience.
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