I have written a bit about the various scholarly views on this here.
In brief:
The "orthodox" argument is that they thought one would show we had it, two would show we had as many as we wanted. The idea is that the Japanese would think we could only produce one bomb, but we were demonstrating we had an entire production assembly line. (The is sometimes listed as a "bluff," as an aside, as if we did not have more bombs in the pipeline. But we did — the core for next one was being finalized and could have been used around August 17th if the war had gone on and it had been authorized. They projected having three bombs per month thereafter and even speculated as to how they might use them.)
There is also the argument that each bomb justified one of the two major production plants (Oak Ridge, Hanford) because they used different fuels. That is, each of the production sites was an expensive, controversial gamble, and the people in charge of making the bomb were very worried about whether their gambles would be considered prudent once the war ended. Using two bombs, in this argument, was about proving the correctness of their choices. (One that was, if true, successful, since nobody ever questioned the redundancies after the fact.)
There is also the argument that they ended up using two just because they had two (coincidentally) ready at the same time, and the people in charge of making this kind of decision were viewing them as just operational weapons (not "special" ones), and didn't think to ask whether they ought to use "just one" or not. This argument suggests that efforts to "rationalize" the narrative after the fact overlook the chaos and uncertainty that reigned during the period in which these decisions were made, and suggests that the people making these decisions were not the political leaders but the military and scientific staff "on the ground."
Separately I would point out that:
the very short delay between the first and the second was caused by weather conditions, not strategy. In the original plan, the first bomb would be dropped on August 3rd, the second around August 10th. So there would have been an entire week in between. Bad weather pushed the first bomb to August 6th, and forecasts caused them to move the second one up to August 9th, leaving a gap of only 3 days – not a lot of time for the Japanese, who needed to assess the damage and reality of the bomb and then come to a major political decision as to whether it would alter their course in the war.
I have not found any evidence that Truman knew a second bomb would be ready as soon as it was. He asked about the "schedule" of bombs in late July 1945, when they were preparing the strike order (which gave detailed information on how the first bomb would be used, but then said that the military on the ground would basically make decisions about any future bombs unless told otherwise). The telegram that was shown to him read only: "First one of tested type should be ready at Pacific base about 6 August. Second one ready about 24 August." This omits mention of the non-tested Hiroshima bomb type — it is a discussion only of the plutonium implosion bomb of the sort tested at Trinity and dropped on Nagasaki. Did someone orally tell Truman there was another type of bomb ready? I have not seen evidence of it — Truman's diary discussions about the bomb are exclusively about the tested bomb at Trinity, and there is nothing else in the documentary record to indicate he knew another one would be so close behind. On August 10th, Truman ordered that the military stop dropping atomic bombs on Japan without his express authorization — what I interpret as an effort to reassert control over the entire enterprise. All of which lends more credence, in my view, to the "it wasn't much of a deliberate strategy" view of why they used two bombs.
After the fact, all of the main parties involved told the "orthodox" narrative, that everything was planned ahead of time with great care and went smoothly. But the documentary record shows a much more chaotic, complicated situation.