* This was from the same period in which the CIA attempted to assassinate Fidel Castro with exploding cigars, and administered a couple of grams (not micrograms or milligrams!) of LSD to an elephant, just to see what it did. They were, not to put too fine a point on it, thrashing around without oversight. (Ranelagh's book, cited in the Wikipedia references, is well worth reading for more on the subject.)
* They had a real, perceived need to listen in on diplomats' conversations in noisy environments (Washington cocktail parties). Cats have very sensitive directional hearing and are less likely to attract suspicion than parabolic microphones. And the fifties and sixties were the heyday of interest in operant conditioning and brainwashing in humans -- and the use of techniques like Skinner boxes for training animals in quite complex behaviour patterns.
In other words: we may point and laugh today, but at the time (a) the Company was so far adrift from reality that this wasn't particularly weird by CIA standards, and (b) it's a classic case of "we need to accomplish (a)", "(b) is a possible way of accomplishing (a)", "okay, let's throw all our resources at (b) (however outlandish and weird it might sound to a sane by-stander)".