To begin to speak; to start talking (to open up or begin a topic).
子どもが急に話し出して、みんなが驚いた。
The child suddenly started talking, and everyone was surprised.
彼女はその件について聞かれると、ためらわずに話し出すタイプだ。
When asked about that matter, she is the type to start talking without hesitation.
ミーティングで課長が新しい方針について話し出した。
In the meeting, the section chief began to talk about the new policy.