。
Breaking a promise — how irresponsible!
Literal
Promise [obj-を] break-[exclam-なんて] irresponsible [copula-だ].
~なんて adds an exclamatory/dismissive shade — 'such a thing as X,' often translated as 'how X!' or 'the nerve to X.' It's a topic marker of contempt or disbelief. Contrasts with neutral ~のは and formal ~とは. The bare 無責任だ closes with the plain copula, giving a punchy final judgment.