Did you ever hear the tragedy of Darth Plagueis the Wise? Darth Plagueis was a dark lord of the Sith, so powerful and so wise, that he could influence the midichlorians to create...life. He had such such a knowledge of the dark side, he could even keep the ones he cared about from dying.
The implication is that Palpatine used Sith secrets that he learned from his master, Darth Plagueis. When confronted by Kylo Ren about his survival, he claims that:
PALPATINE: The dark side of the Force is a pathway to many abilities some consider to be unnatural.
Presumably, this echoes his pronouncement in Revenge of the Sith about the ability to cheat death.
As to the nature of those abilities, in the scene where the Resistance discuss how he survived, one of them speculates about "dark science, cloning, secrets only the Sith know." Although, in-universe, they have no means of knowing what was behind Palpatine's survival, this provides an out-of-universe clue.
We see a further clue upon seeing that Palpatine had several copies of Snoke in cloning vats. This shows that he does in fact have a means of cloning, at the very least, suitable bodies for Force-sensitive people.
Further, Palpatine says that he did die, and it doesn't seem likely that even he could bodily survive the explosion of something like the Death Star if he were at the center of the explosion, not to mention floating in space for who knows how long afterwards. His body would be ash. And, again, reconstituting his body would be a Force power beyond what we have seen so far.
Further, he says to Rey "Kill me, and my spirit will pass to you." So Palpatine at this point is clearly capable of pulling shenanigans with transferring his soul.
All of this points to Palpatine having survived in a very similar way to the Legends universe. In Legends, he had several clone bodies, which his dark side essence would possess. His power caused them to deteriorate ahead of time, though, which when combined with the usual accelerated clone aging would certainly explain his appearance in the film.
One point against this idea: In the film, Palpatine is on life support and his hands show serious injuries, possibly consistent with being unable to fully protect himself from the burns caused by the Death Star explosion, and not so much with the cloning idea. However, his face seems almost the same as it was previously, except for what appears to be glaucoma, and one would probably expect much more serious damage.