This event (Truman being chosen as FDR's 1944 running mate during the Democratic National Convention) is one of the great political stories of the 20th century. The person chosen to run as Vice President in 1944 was a fundamental issue as FDR's health was seriously declining. His next running mate would very likely be the next President as well. The current VP, Henry Wallace, was incredibly unpopular with many of the leaders of the Democratic Party as he was considered too liberal and unreliable. Conversely, however, Wallace was popular with the convention delegates.
First of all, Truman was hardly the consensus candidate for Wallace's replacement in 1944. The convention began with Wallace seemingly having the nomination at hand, while Truman had less than 5% of the support. Party leaders quickly tried to rally support for Truman on the convention floor. Truman was favored by party leaders as a compromise candidate. He supported the administration on most issues, was acceptable to the unions, and he had opposed Roosevelt's reelection to a third term, which pleased conservative anti-Roosevelt Democrats. He had supported Roosevelt's foreign policy while remaining close to isolationists in the Senate. Truman had also been an early supporter of the New Deal.