WASHINGTON—The House Intelligence Committee on Friday voted to make public the transcripts of more than 50 interviews conducted as part of a yearlong investigation into Russian interference in the 2016 election—giving the public details of what key players in the drama told investigators behind closed doors.

The transcripts to be eventually released include some of the people closest to President Trump, including his son Donald Trump Jr., his son-in law Jared Kushner, his former attorney Michael Cohen and his former political...