How did Starbuck come back?

I don't remember there being a solid explanation for her new state/resurrection. An astral projection is one possibility, but I'd like to go even further to explain her pivoting role regarding the location of earth. It could be she was some sort of angel, sent out to guide the humans, as she disappears in the end when she has fulfilled her purpose. She is quite heavily involved in the faith/god storylines so I always assumed her resurrection to be faith related. Something brought her back in a grand scheme.

Her being a cyclon is a bit harder to believe as no real supporting evidence is given (but then again, there is no real explanation) and it would not really fit to here personality. It would also not explain how she vanishes in the end.

From both the TV series and tie-in "Final Five" Comic mini-series, we can deduce that Kara "Starbuck" Thrace is a kind of "angel" who is one counter part of two, the other, Pythia (Saul's Mother, Final Five Comic Reveal), whom together make up Aurora, Goddess of the Dawn.

Sometimes, A Great Notion!

The series, like a lot of science fiction & fantasy, features a spectrum of characters with metaphysical properties to convey the sense that anything is possible and everyone has "a role to play". There are then different "types" of phenomenology to help play into the notion that there IS something BIGGER than the self at work, leading the characters to their fates. None have all the power, which then adds an air of mystery to the work. Battlestar [re-imaged] is basically a metaphysical Space Opera, so of course it leans heavy on "God" and notions about apotheosis.

Head Six and Head Baltar are other kinds of angels, but more specifically seem like guides or "Messengers". It's unclear if they are solely astral projected, but Starbuck clearly is more corporeal upon returning from her disappearance, setting her a bit apart.

Lacy Rand: Trinity, that's what you are.

The Prequel TV Show, Caprica further fed into this by fixating on [technological] apotheosis. Zoe, for instance becomes essentially a holy trinity, as she manifest into a holographic spirit in V-World, The Robot Centurion/"Skin Job", and then Messenger Zoe also exists to help guide the original Zoe and the Holographic Zoe.

Eventually Zoe and Tamara Adama (Joseph Adama's Daughter also killed in the bomb blast) somehow can "cheat" death in V-World seemingly because their holographic counterparts are apart the originals' souls and therefor can not die within the holographic space.

They become "The Avenging Angels" in the New Caprica City V-World Game.

This additional information is important is because,

1. Dr. Gaius Baltar and Caprica Six also were in a bomb blast like Zoe and it is their 3 images that are represented by The Messengers, even though, like Zoe, we know the image of the messengers can exist prior to the corporeal counterpart's death or birth. Head/Messenger Six helped aid Michael Tigh (Saul's father) and Pythia on Kobol when Pythia was imprisoned. (See first comic image above)

2. Because Starbuck's Death has "3" variations. Lee says, he sees her Viper explode in the Nebula. Starbuck claims she went through, went to Earth, and came back, and then the revelation that her burned body and viper were still recognizable and somewhat in tact when the characters visit the first Earth in season 4. The Nebula, a black hole, conveys a notion of playing with space-time, but the fact there are 3 variations in events could feed back into "trinity" allegory.

3. When we look at the life of Zoe, we can gather she was brilliant child whom was to help [re] create centurions in the Caprica/12 Colony pre-fall era, as she was "drawing" them as child. This puts her in line with Hera and her messenger six art and musical notations (furthered by Starbuck), and when you see how Battlestar ends for Hera, one can see a cycle full-filled in the sense that Zoe never got the love & devotion of her parents she deserved, instead she joined a monotheist cult and died for it, while her father Daniel (same name as Kara's father AND the missing #7 that the Cavils destroyed) continued to use her knowledge for profit gains post Zoe's death. Hera is able to spend the remainder of her life with her parents without technology and among simple indigenous Earth 2 peoples. Like Starbuck & Pythia, these two characters, are like counterparts in space-time.

_{Note: Always thought it was curious that Zoe and Hera looked like they could be Caprica 6's and Baltar's children, more than either of the childrens' biological parents, especially when Baltar and Caprica are prophesied to play parental/guardian roles in the final deadly sequence that parallels the Opera House sequence.}

Starbuck and her ship were resurrected, but the same resurrection satellite that resurrected the final 5 after their "Earth" was nuked by their own Cylon uprising. Her true remains, and crashed ship were found on the post nuked earth so it makes perfect sense. Why a human was resurrected? The 5 were indistinguishable from humans, so she got caught up in that construct. The whole "I'm done here" and disappearing while talking to Lee, I can't explain, but they were taking such a metaphysical/Religious slant at that point it was fitting.