As a sound technician I can say that it's pretty easy to do this, even with the technologies available in the 60's. When they make the original recording of a song every voice and every instrument is miced separately and recorded on it's own track. Later on in "remastering" each track can be manipulated or removed, and other tracks can even be added. All you have to do is record a voice saying what you want to put in backwards, then re-record it as a new track with the tape running backwards . It will sound strange and unintelligible, so you simply insert it along with some other sounds that seem strange too. Some of the "psychedelic" rock back then had a lot of strange sounds to hide the backward masking in.
But did they really do this? I don't know. I do remember hearing they did, but never tried turning records backward myself to see if I could hear it. It was the parents of the younger generation making these claims and usually it was simply because they didn't like the "new" music their kids were listening to so they wanted to discredit it. I think a lot of it was the conspiracy theory of the times. It wouldn't surprise me if it had been done sometimes, but if it was, it was probably a lot more rare than people made it out to be, and I wouldn't be surprised if it had a lot less actual effect on people than they made it out to. Afterall, this was the generation of parents that also believed sitting too close to the TV screen would make you go blind from the "radiation" coming out of it!