Q:

Why were people led to believe that the Apollo mission was fake in Interstellar?

In the movie Interstellar, Cooper goes to school where he and Murph's teachers have this conversation.

Teacher: It's an old federal textbook. We've replaced them with the corrected versions.
Cooper: Corrected?
Teacher: Explaining how the Apollo missions were faked to bankrupt the Soviet Union.
Cooper: You don't believe we went to the Moon?
Teacher: I believe it was a brilliant piece of propaganda that the Soviets bankrupted themselves pouring resources into rockets and other useless machines.

So, people were led to believe that the Apollo mission was fake and designed to bankrupt the Soviet Union.  Of course it was a great thing in the history of mankind, but why were they led to believe so in Interstellar?

A:

They were led to believe the moon landing was faked in order to socially engineer the public into living a sustainable lifestyle.
With the Blight having devastated Earth so much that the population is believed to be around 300 million, and food a scarce commodity still, the world governments decided that the consumerist, capitalistic belief systems were not good for society. The general belief that NASA faked the moon landings, along with all the other changes that society has gone through, is used to acclimate the general public into a system of belief of survival of the group over the benefit of the individual.
Multiple times throughout the movie we are told that this is the ideal thought process of the modern man. This is why Cooper's son is pushed towards farming. This is why Murph is ridiculed in school. This is why Professor Brand pushes Coop and Brand Jr. into Plan B through deception. Because that type of thinking moves people away from a sustainable Earth, meaning people die from lack of resources. First the teachers, spouting the propaganda of the day:

If we don't want a repeat of the excess and wastefulness of the 20th century, then...
  ...we need to teach our kids about this planet, not tales of leaving it.

Then Brand:

He knew how hard it would be to get people to work together to save the species instead of themselves.
  Or their children.
You never would've come unless you believed you would save them.
  Evolution has yet to transcend that simple barrier.
We can care deeply, selflessly about those we know, but that empathy rarely extends beyond our line of sight.
He was prepared to destroy his own humanity in order to save the species.

The entire point being that The needs of the many, outweigh the needs of the few, or the one. I mean, the wants and desires of the few are dwarfed by the need for the species to survive.
The push away from technology, from the Moon Landing, was a way to engineer society into thinking about the needs of the many, instead of the wants of the individual.
It was to curtail individual wants and excess. That's why Tom was forced into farming, when his aptitude showed promise that society no longer encouraged in the modern crisis.
Interestingly enough, it's the same reason NASA still exists, in secret.
While it's beneficial for most of society to focus on survival, on extending the human race's existence, it's also important to look for solutions to the problem. Farming is a stop-gap, keeping humanity alive. Expanding into space is the solution, to allow humanity to THRIVE. Politics gets in the way and limits funding, but they were still funded enough to the point we see in the film. And as Brand does, people are tricked into helping because the greater good calls for them to be tricked. A person is smart. People are dumb, panicky dangerous animals, and you know it.

A:

It's a nod to Moon landing conspiracy theories.
You either believe it or not but it's one of the most famous conspiracy theories that the US staged their moon landing to trick the world. Interstellar just used it in the film as a nod to it. Many other films/TV shows also did it. 
Now let's come to your main question.
Why were people lead to believe that the Apollo mission was fake in Interstellar?
Because the society need more farmers not engineers. So no need to push people into dreams of going to space, as clear from Tom's teachers' dialogue:

Okay. Well, right now the world doesn't need more engineers. We
  didn't run out of planes, or television sets. We ran out of food. The
  world needs farmers. Good farmers, like you.

So society needs more people to do farming, NASA already have sufficient people working, no need for inspiring more people to join.
But on the other side Cooper still believes the US landing on the moon was real and fights for it during the discussion. So it also gave us an opportunity to see Cooper's belief system. 

A:

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.

