I get the arguments for this idea. Why did USSR play along? They (claim they) had all the same transmissions and everything else we did, and they called us to congratulate us. I guess they could have just made a wild guess, but it seems more likely that they ... observed us land on the moon before calling us to congratulate us on landing on the moon.
Likewise we observed all their unmanned flights, and our records for all this agree.
USSR did they own hoaxes. Both countries have shit on each other. If one spills the other, it's tit-for-tat and both lose face. Read about Yuri Gagarin and failed US cosmonaut missions and how much they lied about USSR achievements.
There's more to moon and going to outer space than is told. This relates to other species, our limited knowledge/technology, humans not being at the top of the food/IQ/tech chain and the fact that how difficult it is to tell what planet earth is, without losing control
Momentum. Once you get a lie going, it becomes part of organizations, culture and systems. It starts having it's own lies. At some point most newcomers to the system don't even know they are still propagating a lie (because they don't stop to think critically and weight the evidence from a clean slate).
Information asymmetry (I know, you don't) is the key to power, even more so than money (which can be made through information asymmetry. US knows some, USSR knows others. Both share some, but keep other bits hidden. Both want to keep this info from their own population. If they spill USSR data, USSR spills their data and they lose the info asymmetry over their own population.
Whatever USA/USSR know, is not under control and they want both to keep things under wrap, until they can control it more or at least appear to have the illusion that they control it (illusion of control is required for the position of leadership : you lose control, you are not leader material anymore).
Many many potential reasons. I just laid out some of the most obvious.