The speaker means that they fly very fast and efficiently, as if they are experts or professionals at it.
In the given context, the speaker is using a metaphorical expression to describe how efficiently or effortlessly someone or something is able to fly away. The phrase "like nobody's business" is an idiom used to emphasize the speed or proficiency at which something is done. Therefore, when the speaker says "they fly out...like nobody's business," they are suggesting that the subject being referred to (which is not specified in the question) flies away quickly and smoothly as if it were extremely proficient at doing so.
