The US program went from being an exploratory, investigative program to being a bomb production program in late-1942/early 1943. This involved turning over the bulk of the work to the US Army Corps of Engineers. General Leslie Groves, who was put in charge of the Manhattan Project, took a very aggressive approach. First, he made sure that the project got the highest possible war priority rating, so he could acquire any people or resources or money he needed to make it work. Then he pursued all feasible methods of producing a bomb simultaneously, because he didn't know which would be the ones that paid off. (Several of them paid off, in the end, though some did not.) He created close collaboration between the military, the scientists, and the industrial contractors who actually built and operated the fissile material production plants. He moved the most sensitive scientific work (e.g. bomb design) to an isolated location and let the top scientists put pretty much anyone they wanted on the staff. He made sure the scientists were themselves motivated, and picked an excellent (though unexpected) scientific director, J. Robert Oppenheimer, to coordinate the scientific work. Oppenheimer's talents and personality insured that Oppenheimer himself was willing to make almost any sacrifice to get the job done, and he motivated those around him to feel similarly. Everyone loved Oppenheimer.

Ultimately they made a bomb in the time span that they did because they threw a lot of money, science, and labor at it. The project employed some 600,000 people over the course of the war. It spent $2 billion dollars building a new industry from scratch. It tried to guaranteed against failure by producing everything redundantly. Groves went "all in" on it and pushed it as hard as he could.

In terms of secrecy, he made it so that nobody (except those at Los Alamos) knew the whole story, and he gave his counter-intelligence forces a very wide latitude. He wasn't totally successful at this, mind you. The Soviets cracked the secrecy quite quickly, and if the Germans and Japanese had been looking for evidence of a project, they would have found it quickly. There is only so much you can do to hide a project that involves that many people. He had moderate success in getting the US press not to write about it, but only moderate — there were leaks, there were instances of him having to try to use rather heavy-handed censorship to get what he wanted, there were promises he had to make to news organizations to keep them from blowing the lid off the whole thing. He had struggles with secrecy the entire time; every month there was a crisis that almost led to it being put out there in a major way (like Congressmen who threatened to talk about it on the House floor if they weren't told more about what it was). It was not an easy job and it was not perfectly accomplished, despite the "best secret ever kept" rhetoric that got spread around (by Groves) after the war ended.

Without anyone finding out.

Not true. The program was littered with soviet spies. Stalin knew about the project and its progress atleast as early as 44. Hell he knew before truman did, who only found out when he became president a few months before the bomb's use. The soviets also created their bomb in 49 which was based on stolen US documents provided by the rosenthals.

As for why it was successful. Here are a few things. We are huge. If the project is located in kansas, that means no one during ww2 could bomb it. All other countries were in bombing range of us, based on our airfields, but nobody could bomb us. also we have the means and the man power to pull it off. Our economy was now bouncing up as war production ramped up. We had access to some really good scientific minds. Also it helped that a lot of smart jewish scientist and non jews that were repulsed by hitler got out of europe as hitler was coming to power. This may or may not have helped us, but it definitely hurt the Nazis in terms of brain drain.

Our ability to move material around our entire country without worrying about invasion or air raids helped a lot as well.