That's what happens when we forget who we are and what we achieved.
While the existing answers do a really good job of explaining the in-universe reasons for the government to deny the moon landings as well as tieing it to the common pop-culture myth of faked moon landings, the whole issue ties into a bigger theme of the film, namely that humanity has forgotten what it means to strive for more and to progress and explore.
The whole theme of space exploration being a heroic adventure for pioneers to advance society by exploration and technology is a major theme of the film. In this regard it draws as much inspiration from The Right Stuff as it draws from 2001 in other parts. This motif is reflected in various parts of the film, from the heavy use NASA and its insignia and the reminiscence of the Apollo missions in the Earth lift-off scene to the generally optimistic attitude of the whole film and the ultimate realization that it is noone else than an advanced and progressed version of humanity that saves itself.
But most prominently it's reflected in the character of Cooper himself. He is exactly representing this pioneering attitude as one of the few people who are not ready to give up humanity's progress and accustom himself to the situation rather than actively trying to advance beyond it, as also reflected in his dialogue to Donald after the meeting with the teachers:
It's like we've forgotten who we are, Donald. Explorers, pioneers, not caretakers...Well, we used to look up in the sky and wonder at our place in the stars. Now we just look down and worry about our place in the dirt.
This is also reinforced by the screenwriters Christopher and Jonathan Nolan themselves in an interview with Jordan Goldberg, as printed in Interstellar: The Complete Screenplay with Selected Storyboards where they even go as far as setting it in relation to our current attitude towards space exploration, putting the movie as a parable to remotivate humanity's drive towards the stars (emphasis mine):
JN ...I wanted to do something that reflected what I thought was the current state of human ambition. Which it is to say we congratulate ourselves every day on living in this spectacular moment of technological advancement and progress [...] but we're not going into space. Measured purely by altitude, the human race peaked fifty years ago.
JG ...it does emphasize the importance of the human impulse to explore, and it hits it pretty hard...
CN ...because it keeps an optimism to things, even though there's a negative view of 'where are we now?' But there's a real optimism to exploration...
CN ...Part of the problem of where the negativity comes from has happened in the last twenty years with people trying to look in a more cynical way under the surface about motivations for why things happen. So, there's a very, very powerful belief that innovation comes from war, or human conflict. You know, we only went into space to mess with the Russians or whatever it is. And you go, 'Okay. There are interesting theories behind that. There are interesting undercurrents that, yes, in the past, might have been overlooked.' But now people aren't seeing the wood for the trees. We went to space because it was a cool thing to do. [...] Who will be the most prominent person to be remembered for the last hundred years in the way that historical figures are? And if you ask people that question, you know, they'll say things like Steve Jobs, or whatever. And then you go, no, it's Neil Armstrong! [...] And how can it not be Neil Armstrong? If you look five hundred years in the future, it's like, the first guy who left the earth and landed somewhere else.
JN ...the safe bet was in a million years that alien anthropologists would come to Earth and they would find a stick with a piece of polyester on the moon, and they would say, 'Wow, they almost made it. They got that far.' So, you wash away all the day-to-day stuff that we get caught up in. [...] That drive to get out, to explore the universe, will be the residue that's left behind. Armstrong will be the person that people talk about.
Which brings us back to the "faked" moonlandings. They are a perfect way to highlight this contrast between Cooper and the society he lives in, seeing how the Apollo missions were the pinnacle of human-led space exploration and to this day remain a signpost achievement in this regard. For humanity to ignore and deny that achievement, they ignore and inhibit their own progress.