There are at least 5 valid reasons given; to avoid a land invasion of Japan, to justify the cost of the Manhattan Project, to send a message to the USSR, to get revenge for Pearl Harbour and to end the war quickly. Most people say it was some sort of combination. So did any first hand source that was involved in the decision making ever give reasons for it? Whether it be from politics, the military or science.
The problem with answering this is that there was less unified "decision-making" than most people assume. There wasn't a big sit-down discussion about whether they should use the atomic bomb or not. There were lots of little discussions about how it ought to be used (against a city, or a military base? against Kyoto, or Hiroshima? how should the timing go? should they warn ahead of time? should they tell the Soviets? should it be dropped in the daytime, or at night? what should be said after it is used?), but no central discussion about whether it should be used.
As a result you have a lot of people who had different and sometimes motivations participating in the process. So James Byrnes, the Secretary of State, was in the camp of "end the war" and "send a message to the USSR." General Groves was in the camp of "justify the cost" and "contribute to the end of the war" (he did not think one or even two would end it). Robert Oppenheimer was maybe the same as Groves but also "scare the world into enacting a treaty to control nuclear weapons" as well (a very different sort of motivation). And Truman himself seems to have been in several camps, though he rarely gave much indication — it wasn't a thing he felt the need to justify, it was an obvious thing to be in favor of doing.
A separate question one can raise is: how much did each of these people understand about the atomic bomb? (Groves and Oppenheimer: a lot, at least about its technical effects and the targeting. Truman: not very much, about either.) Which additionally complicates things.
We sometimes call this kind of thing "overdetermined": the people in charge had many reasons to be in favor of it, almost no reasons to be against it. There were some dissenting voices but they were kept very far away from any real power. So there was not much of an articulation of the reasons publicly, in the same way that I don't have to explain why I eat lunch to anybody — it's not up for debate.
There isn't a single answer. Certainly some of these can be shown to have been voiced by people at the time, and several can be shown to have been voiced after the fact. But there's no reason to expect there to have been one reason, other than the underlying one that they had a new weapon, they were at war, why wouldn't they have used it?