The idea of saving the appearances, or saving the phenomena, is that deep explanations should explain superficial explanations. For instance, scientific models of the movements of planets should be able to model and predict the movements of the real planets.
The same idea ought to apply to religious 'models' includings all the various explanations in scripture, doctrine, and formal theology - yet it is surprising how often they don't.
Consider reincarnation. Reincarnation - of widely varying detailed explanations - seems to be a spontaneous, natural and universal human belief except where is is contradicted by culture - in other words, all tribal religions (animistic, totemistic), plus many Eastern religions in the spectrum of Hinduism and Buddhism include reincarnation.
I infer that reincarnation (in its various guises) is an attempt to save these appearances - to explain, model, systematize some basic human experience or intuition. In some way people experience and feel something (this is 'the appearance') that leads them to create various models of reincarnation that 'save' (explain) these experiences and feelings.
It is interesting that mainstream Christianity did not, and did not attempt to, save the appearances when it came to reincarnation - it simply stated that reincarnation was not true, did not happen - and implied that any of the feelings or experiences that led to so many millions of people positing reincarnation as an explanation for them, were merely some kind of delusion.
What were these 'appearnaces'? I think the basic one is the feeling that 'my life did not begin at my birth (or ceonception)'.
Most people do not have any specific (certainly not any detailed) memories of a previous life or lives; but many people do have a sense that their 'current' mortal life was not the beginning of life for them - there was 'something' of themselves, some kind of essence (spirit, soul or incarnate body or whatever) existing before their conception or birth. Some already existing entity which took-on a body and became an incarnate mortal.
In sum, I think this feeling of some past existence is the basis of theories of reincarnation.
I suspect that this rather vague, but pretty solid, feeling is the basis of the theoretical elaborations of reincarnation.
However, I do not think there is any similar intuition about future reincarnations and their nature and purpose - and indeed these explanations are widely variable between religions; nor do I believe that the sense of having been specifically 'incarnated' i.e. having had a different body - is a part of spontaneous experience.
One strength of Mormonism is that it does explain the most basic intuitions, feelings, experiences concerning a previous life - although the link between the intuitions of Mankind and the doctrinces of Mormonism is seldom made. But Mormonism can say - yes, your feelings are valid; and we can explain them; but not by reincarnation, instead by pre-mortal spirit existence.
This is saving the appearances.