The book is from 1942, so is dated in thought and is linguistically frilly.
What she was getting at here, I think, is that when we think of a baby learning to speak, and it is learning the word "mama." Probably most of the time it will come to speak the word "mama" were times that before were purely emotional. That is mother has a certain emotional valence as an object and at specific times will have even more emotional significance to the baby. Langer is exploring how language takes over for those feelings and will sit on top of those feelings (and goes on to shape and reshape and limit those feelings). "Mama" comes to represent mother, which before the word becomes accessible was some kind of blind object-feeling-sense data bundle. She claims earlier that the baby's ~"thoughts" are very synaesthetic, sense-data is muddled, something she takes from the James view.
"Powers of symbolic transformation" just means the ability to use language, but this she claims is something that takes time. When we come to be skillful masters of language we have long separated the emotional raw feelings from the word-object relationship in itself, though our words are always very much tinged by those earlier feelings (she presents a high brow picture where only the thoroughly educated reach the best forms of these states, i.e. philosophers).
Her point: "Speech is through and through symbolic; and only sometimes signific."
Speech (and a good "mind") is for abstract manipulation of concepts that help us understand the world, and to accomplish this we must painstaking remove emotions and feelings that attach to our words, e.g. the sternness of chairs.
To finish it: "To project feelings into outer objects is the first way of symbolizing, and thus of conceiving those feelings. This activity belongs to about the earliest period of childhood that memory can recover. The conception of 'self,' which is usually thought to mark the beginning of actual memory, may possibly depend on this process of symbolically epitomizing our feelings."
"A mind to which the stern character of an armchair is more immediately apparent than its use or its position in the room, is over-sensitive to expressive forms. It grasps analogies that a riper experience would reject as absurd. It fuses sensa that practical thinking must keep apart. Yet it is just this crazy play of associations, this uncritical fusion of impressions, that exercises the powers of symbolic transformation."
Susanne Langer, Philosophy in a New Key
Do you wish to know more about human beings? Then postulate less.