。
She was eager to show off her newborn baby.
Literal
She [topic-は] born baby [object-を] want-to-show [and-て] was-fretting.
見せたくて (wanting to show) is the te-form of 見せたい (want to show), connecting to やきもきしていた (was fretting/fidgeting with anticipation). やきもきする is a 擬態語-derived verb meaning 'to fret' or 'to be restlessly anxious' — here, with positive excitement rather than worry. 生まれた赤ん坊 uses a relative clause: 'the baby that was born.'