My mother was busy ironing the wrinkles out of the dress shirts.

Literal

Mother [topic-は] dress-shirt [possessive-の] wrinkles [object-を] iron [with-で] smooth-out [nominalizer-の] [with-に] was-busy

しわをアイロンでのばす = 'to smooth out wrinkles with an iron.' のばす = to stretch/smooth out. いそがしかった = hiragana spelling of 忙しかった. ワイシャツ = dress shirt (from 'white shirt').