When we are conscious of a babble of noise within a dream it suggests that there is too much information for us to be able to sort out what is relevant.
The significance comes from the story of the tower of babel when communication is said to have become impossible. Spiritually, babble has the added significance of perhaps ‘speaking in tongues’ – receiving information from a hidden source.
Psychological / emotional perspective: Confusion in dreams usually alerts us to the need to categorize or sort out those things we do not understand. When we ourselves are babbling, we are not communicating properly; when others babble, we are not taking in information properly in everyday life.
Material aspects: As babies are learning to speak, they tend to babble as first experiments, as though trying out language. Babbling in dreams has this same connotation. We perhaps need to speak out more clearly. Talking in your sleep seems to be the mind’s way of expressing suppressed material, or concerns that you have not been able to deal with properly.