They didn't. The justification behind the atomic bombings in Hiroshima and Nagasaki are murky at best. On one hand the use of the bombs is pragmatic. They ended the war quickly, avoided further military and civilian casualties on a larger scale and likely avoided a situation where Japan is split between north and south in the event of a Soviet invasion. On the other hand, the use of the bombs was absolutely intended to justify the sheer costs of the program and to set an example to the rest of the world. On top of that japan had offered surrender (although their conditions didn't meet US terms.). There's no real way to say whether or not the bombing was justified.