They picked a spot that was already on a government "proving ground" (bombing/explosives test site), and after the fact they lied through their teeth about what had happened, sending out press releases that said it was an ammunition dump that exploded. An exploding ammunition dump, as an aside, is a big deal: you can easily have a few thousand tons of TNT involved in such a thing, and indeed explosions of this sort are among the largest non-nuclear detonations experienced, even though they are typically 10 to 100X less powerful than the bomb detonated at Trinity.

The Manhattan Project also had security agents in nearby towns to monitor how people were talking about it; they found that most bought the ammo dump explanation.

So many people did know that it had happened, but they did not know exactly what it was. Some people later reported thinking it was strange, because the brightness of the explosion was much more than a conventional weapon ought to be, and it hung around a lot longer than an ammunition dump explosion, but they didn't investigate or push their own theories. After all, in a world where atomic bombs didn't yet publicly exist, what else could it have been? In any case, the secrecy only had to hold for a few weeks, before the attack on Hiroshima.