There were a couple of reasons why the USA decided to use nuclear bombs against Japan.
The first is that Japan was not prepared to surrender, even though they were clearly beaten. The fighting on Okinawa showed that to continue to use troops to push onto the main islands of Japan would cost many thousands of allied deaths and casualties. This may have become unacceptable to the population back home and may have caused an anti-war sentiment. People were becoming war weary and as the war in Europe had already ended, a quick solution to the war with Japan was desirable.

Great Britain had already showed that continual bombing raids against civillian population centers did not lessen the resolve of the population to continue the fight but did indeed have the opposite affect of strengthening the resolve and willingness to continue. The solution had to be something that was so devestating and so shocking that it would break that resolve quickly. Nuclear strikes were seen as perhaps having that kind of affect.

The second reason was that Russia was known to have captured certain German scientists that had been involved in the German nuclear program and it was considered expedient to firstly develop a nuclear weapon and then show that the USA would be prepared to use it against an enemy. Japan just happened to be the enemy at the time and the situation I described in the first section made the decision to act somewhat easier to reach.

Should the USA have used those weapons? I don't want to answer that as I cannot put myself into the position of those who had the heavy burden of making that decision. It has been said that (possibly) millions of lives were saved by doing so, in a war that was basically over, except for the refusal of the Japanese to surrender. I am not sure if that is enough reason to excuse the use of a nuclear weapon as there may have been other ways to bring the war to a speedy end that were overlooked at the time because of focus on the nuclear option. I just don't want to make a judgement on it at all.