Smiling sadly, she began to speak.

Literal

Sad-looking smiled [while-ながら], she [topic-は] talking began.

悲しそうに微笑みながら combines ~そうに (appearing sad), 微笑む (to smile gently), and ~ながら (while doing). 話し始めた uses ~始める ('to begin'). A rich literary image with simultaneous actions.