Doesn’t the Senate have to start such a proceeding?
No. The House will bring up and vote on the articles of impeachemnt, i.e. the charges against the president. If a majority of the House votes to impeach, the articles move to the Senate where a trial of sorts is held with the Chief Justice of the Supreme Court presiding. If 2/3 of the Senators present (assuming they have a quorum, i.e. 51 members present) vote “guilty”, the president is immediately removed from office and the VP is sworn in as president.