“Then explain how the Russians could send probes to the Moon but not a manned mission.”
On September 13, 1959 the Soviets demonstrated the technical capability to crash something into the lunar surface. On April 12, 1961 the Soviets demonstrated the technical capability of putting a human being into space and having him not die for a couple of hours. So, at some point after April 12, 1961, the Soviets achieved the technical capability of shooting a cosmonaut at the Moon and smashing him into the lunar surface. Of course, even the godless Commies wouldn’t actually do that, because it would have been crazy.
On February 3, 1966 the Soviets demonstrated the technical capability of landing a cosmonaut on the Moon and leaving him there to die.
It was not until September 12, 1970, that the Soviets demonstrated even the theoretical capability of landing a cosmonaut on the Moon and returning him safely to the Motherland, at which point it was “too late” as the Americans had “beaten” them over a year ago.
Of course I’m glossing over issues of payload size. (And the length of time it was possible to keep people alive in space, but the Soviets succeeded in keeping someone alive in space for nearly five days as early as 1963, and a manned lunar mission can be done in a little over eight days, so not a huge stretch. I do think that the life support and logistical issues of a human spaceflight to Mars–which would probably take years, rather than days–would be way more challenging, even if we had the capability of sending things to Mars and then bringing those things, or at least parts of those things, back to Earth, which so far we don’t.) But I do think MarvinKitFox is basically right: For lunar missions at least, the differences between manned and unmanned missions aren’t that large, boiling down to reliability (the first time the Soviets tried to crash something into the Moon, they missed the Moon entirely; Luna 16 wasn’t the first attempt at an unmanned sample return mission, either) and the added difficulty of going there and then coming back. Basically that whole pesky “…returning him safely to the Earth” part which does make manned missions more challenging.
But the point is, Conspiracy Theories to the contrary, there isn’t some kind of Instant Death Radiation Belt beyond Earth’s atmosphere (or beyond Low Earth Orbit), or any other particular factor that makes manned lunar missions impossible with 1960s technology. We (the human race: Americans and Soviets) demonstrably had the capability to keep humans alive in space for at least days at a time. We had the demonstrated capability to land things on the Moon, with increasing precision (and it didn’t take us months or years to get there either, so it was well within our demonstrated capability for keeping people alive in space). We had the demonstrated capability of landing things on the Moon, and then of bringing stuff back from the Moon. Apollo was not impossible. Q.E. fucking D.