Since 1945, the US decision to drop atomic bombs on Hiroshima and Nagasaki has been the subject of tremendous controversy. However, at the time, there was little controversy over it. The bombs were essentially seen as another tool of war that could and should be used to bring an end to the war as quickly as possible.
Since 1945, people have suggested that the US had ulterior motives for using the bomb and that it should have tried other options. People suggest that the US dropped the bomb so as to scare the Soviets and to make them realize that they could be harmed badly if they became too aggressive after the war. People suggest that the US could have demonstrated the bombs on uninhabited islands or otherwise gotten Japan to surrender without actual use of the bombs.
However, at the time of the dropping of the bombs, none of these was even considered. Winston Churchill argues that there was essentially no decision to be made. It was, he said, just assumed that the bombs would be used when they were available just as any other weapon would have been used. It was very clear that an invasion of Japan (the next most likely outcome if the bombs were not used), would cause the deaths of huge numbers of Americans (not to mention Japanese). With the war having dragged on for a very long time, and with the Japanese having fought tenaciously against invasions of such places as Saipan, Iwo Jima, and Okinawa, it seemed clear that the thing to do would be to do anything possible to end the war.
The US motive behind dropping the bombs, then, was a desire to end the war as quickly as possible and with as little loss of life as possible.
Perhaps one of the reasons that motivated the Americans to drop the atomic bombs on Japan was the fact that by using the bombs, the US could seek to control the circumstances of post-war Japan and to impose American control in the redesigning of Asia. The sheer costs and effort spent on developing the nuclear warhead during the Manhattan Project also meant that the concentration of cutting-edge scientists and engineers employed were generating a momentum that could not be stopped - they wanted to see the atomic bombs in use and Japan provided the best opportunity, at that time, to do so. Casualties were also a key factor. Truman had been informed by his military advisers that it was necessary to invade the Japanese mainland since the imperial army would fight on a suicidal scale to counter the American troops. Due to such a tenacity in fighting, a high casualty count was to be expected from the American forces. This estimation greatly influenced Truman in his decision-making, since the operationally-ready nuclear weapons could be used to force Japan to surrender without an invasion of the mainland and the incursion of high casualty numbers - they could be used to preserve American life.