In Star Wars Episode IX: The Rise of Skywalker we see Force heal used at multiple points by Light Side users. However in Star Wars Episode III: Revenge of the Sith it seems that this is a Dark Side exclusive power. Anakin doesn't know how to use it and Palpatine says that it's not a power you can learn from a Jedi. What's the explanation for this?
The power that Darth Plagueis had was distinct from what was used in The Rise of Skywalker.
In TROS Rey and Ben use the Force to transfer their life between themselves and other organics, there is no new life being created, this is why Ben has to die to save Rey. On the other hand, Palpatine says that Darth Plagueis influenced the midichlorains to create life, ie he wasn't transferring life, he was creating new life.
Anakin probably could have transferred his own life to Padme but then he would have died instead, but if he could create life the way that Plagueis could then they could both survive.
Palpatine says that it's not a power you can learn from a Jedi. What's the explanation for this?
Palpatine was likely trying to manipulate Anakin. Throughout the prequel trilogy, clone wars,...etc Palpatine is constantly manipulating different people to achieve his goals. While it is possible that the power he described exists, he is more likely claiming its existence in an attempt to manipulate Anakin.
I’m not sure that any of the Jedi during Anakin’s time knew about Force heal. We never see it used during the Clone Wars, a time where that ability would be very useful. I think it is possible Rey learned it from the Jedi texts. Then when she healed Ben, he learned about the ability and used it to save Rey later. Before you ask, I’m sure Baby Yoda relied on his instinct or the will of the Force to use Force heal.
In the Legends novel Shadows of the Empire, Darth Vader does use Force Healing. In the movie Empire Strikes Back, the time you see him with his helmet off-is actually a chamber he uses not only for communication, but also to focus his time into healing the permanent damage. In the SoE book, he uses as much hate as he can, but can only remain outside his breathing apparatus for something like 15 minutes, and it causes him pain to even do the act-using more rage to make himself last longer each time.
It may just be Legends, but it fits into the canon universe.