The verb "to rain" can be transitive. "To rain X" basically means that "X is falling out of the sky", whatever X is.
For instance, "On Saturn's moon Titan, it rains methane."
"Raining cats and dogs" literally means that small animals are falling out of the sky. But, of course, this image of animals falling from the sky is a metaphor for very large, heavy drops of water (and possibly dark skies, since animals are opaque).
The phrase is not an idiom, as the other answers misinform you. An idiom occurs when some piece of bad grammar is given an accepted meaning ("catch as catch can"), or when some sentence or phrase is given a completely different meaning from what it appears to say at face value ("kick the bucket").
"A and B are as different as X and Y" means that A is different from B, and this situation between A and B is similar to the situation between X and Y, which are also different from each other. The differences between A and B can be similar or parallel to those between X and Y, or they might not be. If A and B are very similar, then we actually expect X and Y to be similar.
"Classical music and grunge rock are as different as day and night". In this case, we learn nothing from the analogy, except that the speaker feels very strongly about how different those two kinds of music are. Two kinds of music are different in ways that do not really correspond to the differences between day and night.
"Porpoises and dolphins are as different as cats and dogs". Here, porpoises and dolphins are actually similar, but they have differences. Cats and dogs are also similar, and have differences. Those differences are somewhat parallel to those between porpoises and dolphins.