It's also important to remember the timing of this act. It came not long after World War II, where, because so many working Americans were in the military and overseas, and the demand for war material so great, that the workers who remained behind, including African-Americans and women, had made significant gains in wages and benefits. Unions enjoyed large memberships and real political power.
Taft-Hartley was an attempt to rollback some of those gains on behalf of wealthy business owners, and right at a time when the economy was rapidly expanding and the middle class was demanding ever larger numbers of consumer goods.
Truman's political party, the Democrats, were very unpopular that year as well.