。
She blushed with embarrassment.
Literal
She [topic-は] embarrassment [by-で] face [object-を] reddened.
恥ずかしさで ('from embarrassment') uses the ~さ suffix to turn the i-adjective 恥ずかしい into a noun — 'embarrassment' as a feeling. で marks it as the cause. 顔を赤らめる ('to redden one's face / to blush') is a literary expression — 赤らめる is the transitive form of 赤らむ ('to turn red'). More vivid than just 顔が赤くなった.