As belief in conspiracy theories is associated with the rejection of science, it is likely that this answer will be rejected by many at the outset. If you doubt the reliability of science, perhaps you should check out this NPR article. Please note that it is an opinion piece. The author Alva Noë is a professor of philosophy at UC Berkeley, not a journalist. Use your own reasoning to honestly consider his words.
If it is not the scientific method itself that you distrust, but the "scientific establishment", then you should consider the way science works. All papers are subject to review, all results are expected to be repeatable and independently verified. Individuals can and have created independent groups to ensure the integrity of their work. If there is not a significant group of scientists with expertise in a particular domain that reject a conclusion from that domain, then it would be prudent to provisionally accept that conclusion.
How, indeed, can we believe that anything we did not personally witness, did in fact happen?
That's a trick question...critical thinkers don't have to "believe" anything. As paleoanthropologist Dr. Briana Pobiner (quoted here by Adam Blankenbicker) replied when asked "You believe in evolution, right?" -
"I don't believe in evolution - I accept the evidence for evolution."
The believing isn’t what makes evolution true or not, it’s that there
is evidence that supports it.
One of the obstacles to belief in any space enterprise is belief in a flat earth. Those who believe in a flat earth are aware of the idea of orbits and of images returned from space of the globe. Because of this, they assert that NASA must have faked the moon landing, as they reason the whole space enterprise must be a conspiracy to begin with.
However, since we accept science, it is easy to dismiss the possibility of a flat earth. First of all, for the dedicated and resourceful, it is possible to fly around the Earth without stopping. This may be prohibitive for the average reader, but as we discussed above, we don't have to personally witness something to accept the evidence for it. If you're interested in the evidence for a round earth, Moriel Schottlender provides a good summary on her site SmarterThanThat.
There are many reasons we know the Apollo missions landed on the Moon and returned. Here is a list summarizing the main ones:
1. The science of space rocketry is sound.
For the layman, a common area of confusion is how rockets work in space with nothing to push against. However, the scientific principle for why this works has been understood for a long time (Newton's third law). For more information see:
Some have heard that there was scientific skepticism prior to the launch of Sputnik, but this was rooted not in doubt of Newton's laws but in concerns of the financial viability of the enterprise:
How widespread was the notion that space travel was impossible prior to the successful launch of Sputnik 1? - History Stack Exchange
2. The science we use to model and build rockets is the same as we use on Earth
The methods we use to model space flight - in particular the chemistry and thermodynamics of rocket engines - are used in every field of engineering.
The way we model the boiling of a kettle allows us to predict and demonstrate that an engine produces thrust in a vacuum. These models are not in question; one might as well claim that the engine of a car cannot function according to physics as claim the same for a rocket engine.
3. It is clear that humans have been to space.
The International Space Station can be viewed by amateurs from the ground. With dedication, astronauts can even be spotted on spacewalks:
How would one go about proving humans have really been to outer space? - Space Exploration Stack Exchange
That said, some are willing to believe we can go to space today, but doubt the possibility of what was accomplished in the Apollo era. That leads us to...
4. There is a wealth of third-party evidence for the Apollo Moon landings.
The Soviet Union, Japan, China, and India have all, independently of NASA, verified the Moon landings. Wikipedia has a convenient collection of this third-party evidence. The Wikipedia article is well-sourced, and thus acts as a repository of information that has been gathered from places such as JAXA's website, an Indian academic journal of research, and Xinhua News Agency. It is beyond belief that there could be a conspiracy this vast to maintain a fake moon landing story. There could not even be a reasonable collective motivation. China aspires to land a manned expedition on the Moon, and they would love to one day have the opportunity to claim that they were truly the first; yet even China affirms the account of the Apollo expeditions.
One key piece of third-party evidence comes from the retroreflectors left on the Moon by three of the Apollo missions (11, 14, and 15) and by two uncrewed Soviet lunar landers. These retroreflectors have been pinged from dozens of sites all over the world; they are exactly where the US and the Soviet Union said their vehicles landed. The existence of the retroreflectors does not prove that the three US missions were crewed, but it does prove that some kind of landings occurred at those three sites.
The above points show that the Apollo program could have been taken place, and that there's lots of evidence that shows that it did take place. Could it have been fake? This leads us to ...
5. The technology needed to fake the Moon landings did not exist at that time.
The Apollo astronauts took an abundance of photographs during their short stays on the Moon. The photographic evidence is very compelling, which compels the conspiracy theorists to try to debunk those photographs. These efforts inevitably fail; the lighting in every single photograph taken from the surface of the Moon is consistent with a bright light source one astronomical unit away and is inconsistent with studio lighting.
Those photographs could of course have been created as computer-generated imagery. CGI has recently been used to recreate some of the iconic photographs taken from the Moon. One of the two photographs below was taken by Neil Armstrong. The other is pure fantasy, generated by a computer.
The technology needed to produce those photographs as computer-generated images is very new (late 2014, to be precise), and even then, it wasn't easy. Such technology did not exist twenty years ago, let alone fifty years ago.