To begin to say; to start talking or to bring up a matter — to broach a subject or start speaking about something.
彼はついに本当のことを言い出した。
He finally began to tell the truth.
子どもが学校に行きたくないと言い出したので、親が相談に乗った。
The child started saying they didn't want to go to school, so the parents talked it over with them.
会議で彼女が新しい案を言い出した。
She brought up a new proposal at the meeting.
To be the first to say or suggest something; to propose doing something (to initiate a suggestion).
旅行の計画は誰も言い出さなかったが、田中さんが最初に言い出した。
No one had suggested the trip, but Tanaka was the first to bring it up.
飲み会で『次は温泉に行こう』と彼女が言い出した。
At the party she suggested, 'Let's go to a hot spring next time.'
言い出しにくいことでも、誰かが言い出してくれると助かる。
Even difficult topics, when someone starts them, it helps the rest.