The children's cheerful laughter spread through the forest.

Literal

Child [possessive-の] bright laughing-voice [subject-が] forest [inside-の中に] spread.

明るい笑い声 — 明るい here means 'cheerful, bright' applied to sound, not physical brightness. 笑い声 ('laughing voice') is a compound noun. 広がる is intransitive 'to spread' (counterpart to 広げる, transitive). の中に ('inside X') marks the place where the spread happened.