When you lose consciousness or sleep, you seemingly skip over some parts of time. To you it seems continuous, you're okay with the lost time. Things are continuous from your point of view, even when not in sync with real time.
Suppose a very talented doctor could disassemble your brain into 5 pieces, keeps them alive in a vat for an hour and reattaches them such that no harm was done. Supposedly even then you would perceive things as continuous. Next he does it again, except this time into 50 pieces. You are still "yourself". Then little by little increasing this into 1000 pieces, a million pieces, all the way to individual atoms and back. Here it might be a bit of a personal choice, but I would say that even then I'm still me and things seem continuous from my perspective (although probably pretty weird experience).
Now let's say that after he cuts your brain to individual atoms, just as a joke he then stores them for a week before putting them back together again just to see your reaction when you realize a whole month has gone by unnoticeably to you.
He tries to do this joke again, except this time for a month, but unfortunately he gets hit by a bus before he has the chance to put you back together. Now quadrillions of years pass, human civilization no longer exists, and your atoms are flying around in space, occasionally recombining as parts of planets or stars, which then are destroyed and scatter and again become something else.
All kinds of crazy civilizations come and go, but one day a particularly bizarre religion emerges on an alien world. They have determined that for the rest of eternity, their task is to collect atoms from all around the universe and then recombine them in every possible way. Sometimes while doing this their own world is destroyed, but another civilization later on has a similar religion that tries a similar thing, and after endless cycles of this one day one of them happens to put together the combination that happens to be exactly you at the instant the doctor disassembled you.
So you wake up, seemingly no time has passed for you, but here you are again feeling great. You never do find out what happens to the doctor and you live many happy years among the hospitable aliens. You become famous on the planet after you reveal your backstory, and the local scientists decide to replicate his experiment on you. They disassemble you to pieces, but unluckily right then a terrorist attack destroys the facility and they don't have the opportunity to put you back together.
Now insane amounts of time pass again. The universe undergoes the big crunch, but it turns out that every gnab gib is followed by another big bang, except each iteration has a slight random element. Unimaginably many such cycles come and go, and eventually one emerges that is exactly like the one where you got taken apart, except this time there is no terrorist attack. You don't notice at all, but huge amounts of time actually have passed and you are in a new universe.
Every time something destroys you, there will be another point in time where it didn't. So now you have immortality. That's a good start for a religion, but let's go further.
Suppose one time you get destroyed, but it happens that your exact configuration happened to be one that the crazy alien permutation religion tried in the recent past. You wake up, but now you're in the past. But from your point it's continuous, you pick up exactly where you were. Hey it's not just the pattern that is the same, but they even happen to be the same atoms.
Now you continue living your life, but to fit in better in the alien world, you decide to undergo some plastic surgery to look more like the locals. The local doctors helpfully even take out a brain tumor they happen to find. Now your body and mind are a bit different from before, but it still feels continuous to you. You take a vacation to another planet, which just happens to be the one where you were living the last time you got destroyed. There you happen to walk past your past self on the street, but don't even notice yourself. The other you does notice you however, since you're a tourist from another planet.
This way during your endless existence, the pattern of your atoms sometimes changes, but it still feels continuous to you. One day you go wild and decide to recreate the movie Benjamin Button just for fun. The local doctors go through successive iterations to make you younger and younger. It's a fun experiment. However just as they've gotten to the baby stage, unfortunately a gamma-ray burst wipes out their world and your baby version gets destroyed.
But luckily after several cycles into the future, a world emerges where a woman happens to give birth to a baby which is just as you were, and even the atoms again happen to be the same (that took a few googol cycles to occur, but it was worth the wait). You are born to the world as this baby, and shrug off your hazy memories of a life before that as just childish fantasies.
In this way as the cycles go on and on, your pattern takes on many forms. Sometimes jumping forwards, sometimes backwards. Sometimes you are a baby, sometimes an alien, sometimes a man dying of old age. Everyone you encounter is actually you. So neat, now our religion has not only eternal life, but rebirth as well. Have those TODOs crossed out.
But it's not really a proper religion without a god, is it? OK, actually unbeknownst to you, in one cycle the universe had turned out such that there is an alien superpower remotely monitoring all life on your planet. They call themselves the Universe Knowers, or U.K. for short. The U.K. monitors everything you do. If they like what you did, if you lived a good life, if you get destroyed they recreate your pattern in hell or heaven, custom-made just for your beliefs (or your money back).
Most of the cycles you happen to be born into a universe with no U.K. in it, but sometimes there is one. The criteria they use to determine your fate depends on the cycle you happen to be in. So there you have it: eternal life, reincarnation and not just one god but every god ever.