"The cat got your tongue?" -- This phrase is a question commonly posed to shy and or quiet children by adults. It is basically a way of asking a child why he or she is so quiet. The origins of the phrase are based on an ancient belief that a cat could steal the breath from sleeping children.
"Get the monkey off your back" -- This phrase refers to getting rid of a recurring problem, generally an addiction of some form or other. The monkey is a metaphor for the addiction. Think of a crazy monkey riding on your back refusing to let go.
" It's raining cats and dogs." -- This is a very well known phrase which indicates that it is raining very hard. This phrase is also metaphoric and was most likely deduced from the fact that, in Germanic mythology, cats were associated with storms and dogs were associated with rain .
These are just a few examples of fairly common animal phrases, the majority of which are based on metaphor. There are also metaphors that are animal based but don't include an animal in the actual phrase; for example, "don't bite the hand that feeds you."