According to Leslie R. Groves in "Now it Can be Told: The Story of the Manhattan Project" the members of Congress who asked about it or wanted to tour facilities were politely but firmly discouraged by the Secretary of War and by February of 1944, Congress was in the habit of issuing more-or-less whatever funds were requested for the project without asking any questions.

Remember that the sum-total of the cost of the project, while astronomical, was about the same as what the country spent in a week-and-a-half on the war writ large. Certainly it was big enough to be on Congress' radar but politically speaking, war has always been more in the executive's domain and Congress was willing to follow the White House's suggestions on the matter.

The project was paid for originally out of a "black budget" (secret discretionary funds Roosevelt had access to). So Congress was not advised, and was explicitly kept out of the loop. As Vannevar Bush, one of the top civilian administrators, told Roosevelt in late 1942 when getting authorization for it:

“It would be ruinous to the essential secrecy to have to defend before an appropriations committee any request for funds for this project and it is therefore recommended that some time in the spring you request the Congress for the needed funds ($315,000,000); such funds to be expended at your discretion.

So in other words, FDR would get some blanket funds to spend, and spend them on this. This is back when they thought the Manhattan Project would cost $500,000,000. It turned out to cost four times that much.

But eventually they needed a lot more money, and were having trouble with Congressmen (including Harry Truman) who were trying to audit various pieces of the project (which they heard about through their constituents). In February 1944 the heads of the project, along with General George Marshall and Secretary of War Stimson, met with 7 Congressmen. These were the Speaker of the House, Senate Chairman, the Majority and Minority Leaders in both chambers, and the Senior Minority Member of the Appropriations Committee in the Senate. They told them a generous amount of information, noting that at one point they had thought they were neck-and-neck with Germany (but no longer really believed that), and that it might prove to end the war "in case of deadlock." The Congressmen agreed they would help maintain secrecy, by making sure their appropriations requests went through unremarked upon, and helping squelch any of their fellow Congressmen who tried to audit or pry (and there were quite a few).

So there was a very small core who knew a bit about the overall purpose, but were not in the know on the ongoing operations. This core helped keep it off limits from the others. It is not unlike our present day approach to big secret programs, where small committees (like the Senate Intelligence Committee) hear the classified information and then steer the budgetary requests through the general system.

The Manhattan Project was an extremely undemocratic program at its core — it was about a small power elite making huge decisions about the present and future. It was certainly necessary to get the results they did. But I always think people should keep this in mind whenever they call for a "Manhattan Project for X" — it is not a "normal" mode of scientific and technological development, and should not be applied too readily!