It has been estimated that over 1,000,000 American troops would have died invading and subduing Japan.
Some people claim that the 1 million estimate was a lie that the U.S. government made up after the fact to justify Truman's decision. However that allegation is patently false since the government had nearly a million Purple Heart medals manufactured in preparation for some of the bloodiest combat expected for 1946 and perhaps 1947. I understand that they were using the same stock of medals all the way into the Vietnam War until there was a redesign.
1.) We didn't have the option of simply isolating the Japanese homeland, for they would starve, and their soldiers were still in other places, causing harm. Moreover, there were Allied soldiers being used as slave labor in the country that would certainly die.
2.) An invasion would kill the vast majority of the citizenry, since even children were being taught to resist unto death. It would have essentially ended Japanese culture and would have been a greater bloodbath than we can imagine. Also, Allied soldiers would have almost certainly been executed under those conditions.
3.) The incendiary attacks killed many more people than the atomic bombs, and they continued in hopes that it would compel the government and the Emperor to surrender to save lives all the way around. The atomic bombs were just the latest technology in that campaign. The psychological power of the atomic bombs evaporating whole cities at a time, brought a rapid end to the war and ended the massive bloodshed on all sides.