Like most things, it's impossible to conclusively prove a negative (i.e. that this didn't happen), but the supposed sources are pretty questionable and the lengths to which things would need to be changed to cover this up rule out any reasonable possibility of this being true.
In reverse order:
4. NASA has lost original recordings of most Apollo 11 mission tapes.
As DJClayworth linked in the comments, NASA not only has all the original Apollo 11 mission tapes, but they've put them online. There were technical problems with some of the recordings, but in NASAs own words:
The equipment onboard the Apollo Command Module that was used to make the recordings was called the Data Storage Equipment (DSE). Its contents were transmitted to the ground periodically during the mission. Also, the Command Module DSE had the capability to record data live during certain periods from the Lunar Module as it flew separately in lunar orbit. The equipment used aboard the Lunar Module to make the recordings was called the Data Storage Electronics Assembly (DSEA). It made recordings onboard the Lunar Module, but the DSEA flown on the Eagle during Apollo 11 malfunctioned. As a result, many of its recordings are barely, if at all, audible, with a constant high-pitched background tone. In the attached database, the recordings that are virtually inaudible are highlighted, but they are available on the web to ensure a complete release of the recordings made during the mission.
I'm sure that it's possible to digitally edit out the background tone, but whether that will leave intelligible recordings, I'm not sure. I suspect that's what they did to get the transcripts, but I haven't looked into that.
It is worth noting that the original higher-quality video from Apollo 11's time on the moon has been lost. What's available now was produced by pointing a NTSC-based camera at a monitor displaying the original SSTV signal. However, the lack of original video is hardly the "most mission tapes" that was originally claimed, and the original audio from the moon landing is still available.
3. Dr. Vladimir Azhazha, a physicist and Professor of Mathematics at Moscow University also confirmed that the message was sent to NASA but was censored (supposedly 'confirmed' by interruptions in radio transmissions)
Assuming that "Moscow University" is actually "Moscow State University" (aka "Lomonosov Moscow State University"), I was unable to find any contact information for "Valdimir Azhazha" on either the physics or math departments' websites. That doesn't rule out the possibility that he used to serve there and no longer does, but the only references to him I could find anywhere involve this and other UFO claims. Like so many other "famous" scientists linked to UFOs, I suspect he never existed, or at least doesn't have the positions that are claimed for him.
That said, what of the claim itself? There's certainly the potential for interruptions in the radio transmissions, but there's no way that NASA could cause the Command Module to be censored in real time. As per the claim with #1, if radio hams were picking up the transmissions, then the only possible way for NASA command to have censored it would be if it were received directly from the astronauts, broadcast back to them with censoring, and then retransmitted to the world. That only leaves the possibility that the astronauts themselves censored the transmission... but they could just have chosen not to say anything rather than say something and cover it up.
In short the claim of "censoring" doesn't work because there is no one who could have censored the very same broadcasts that the quote was reportedly from!
2. Maurice Chatelain, former chief of NASA Communications Systems confirmed that Armstrong had indeed reported seeing two UFOs on the rim of a crater.
This is similar to the second claim, except even easier to address. There is no such thing as "NASA Communications Systems", but Apollo did have a "Chief of Communications and Frequency Management". Unfortunately for this claim, that position was held by Paul A. Price (page 217 of the PDF) for Apollo 11.
As for whether Maurice worked for NASA at all - all I can find are counter-claims that he worked for a NASA sub-contractor and had been fired before Apollo 11 happened. Since I can't find any definitive source either for or against, I have to leave that aspect of this open. However, it is certain he didn't have the claimed position during Apollo 11.
1. Science-fiction writer, Otto Binder claimed that some radio hams received the message from NASA in which Neil Armstrong said that he had seen other spacecraft.
There's a great article here which discusses what it took to overhear the Apollo mission directly, including a contemporary article about it. Noteable quote:
Baysinger's wife and daughter watched the Apollo 11 landing on TV while Baysinger and Rutherford listened via Baysinger's equipment. The signal on the home-built equipment came through approximately 5-10 seconds earlier than the signal on TV. Baysinger figures NASA or the TV network [I assume it was probably CBS] put in a delay in case they needed to edit out anything embarrassing.
The signal the lunar eavesdropping equipment picked up was noisy, but Baysinger says you could hear what was going on. Baysinger made tapes of the transmissions, which he still has. In September 2009 he transferred salvageable sections of the tapes to MP3 format for this project. You can hear them for yourself via the links below.
I asked Baysinger whether he found anything that NASA edited out – comments about things going wrong, the astronauts being loose with their language, or exclamations about meeting aliens! He said no – absolutely everything was transmitted to the public on TV. In fact he said, "that was kind of disappointing". Part of the idea of this project was to hear the unedited "real story", and it turned out there was nothing edited out. Indeed, Rutherford's story (click here for hi-resolution version which you can read) makes no mention of hearing anything unusual.
Of course, it's always possible that Baysinger was "hushed up" by whoever is covering it up (as per claim #3), but given that the article in question was published less than two days after the moon landing, and that no one would have known to hush him up before it was published, it seems pretty unlikely that without an actual recording of the supposed quote surfacing there is anything to this claim.
