The mother is bragging about her daughter's typing skills.

Literal

Mother [topic-は] own [possessive-の] daughter [possessive-の] typing [possessive-の] skill [object-を] is-boasting

自慢している = 'is bragging' (ongoing). 腕前 (うでまえ) = skill/prowess (腕 arm + 前 front). Multiple の chain: 自分の娘のタイプの腕前.