The general, who began the whole ordeal, did what he did, because he believed the Russians were poisoning his water. This idea came to him when he could not sexually perform as he would like, and of course the only logical answer to that was the Commies. This is making fun of the ridiculous paranoia that was surrounding everyone’s mind at the time.
Another example of satire in the film, is when it is decided that certain people are going to have to go underground to protect the human race. As the Americans are discussing this, the Russian ambassador takes out a hidden camera and begins taking pictures of the American plans. Even when the world is ending, he is still thinking about ruling it.
What’s wonderful about Dr. Strangelove is that it works even without the context. I first watched the film four years ago when I knew nothing about the history and I still enjoyed it. The film supplies enough information on its own so that those uneducated on the topic can still understand the satire. However it makes the experience so much more rewarding when outside knowledge is in fact known.