I think the key is that magic is given a logic of its own. It has its own structure and system of rules that seem plausible. There are spells and counter-spells that work like arrows and shields except using a wand. Voldemort's historic attempt to kill Harry Potter fails due to a deeper magic whereby the sacrifice of his mother protected him. A voluntary giving of one's life has got to do something right? Voldemort later fails to kill him because their wands share an inner core. Sharing an inner core must mean something right? Then Voldemort fails to kill Harry twice because he used Harry's blood in regenerating himself and in so doing bound Harry's life to his own. If your life was built on someone else's blood then you can't kill him. And of course Voldemort could protect his own life by putting bits of his soul in horcruxes. You can't die if you can't destroy the soul and if some of your soul is hiding someone else then you can't be killed even if your body is destroyed.
The logic is there, ready for you to immerse yourself in it. That is part of why the books are so addictive. But of course that doesn't mean I truly believe the logic. I temporarily assent to it and enjoy the stories and then go back to my regular life. But what about faiths or religions? Aren't they very similar?
Indeed I think successful religions are the ones who present a plausible logic that people hold onto and don't let go. Christianity is a prime example. In my old days as an evangelist I used to present this logic to potential converts: God originally created the world good and pure (so God is blameless). People rebelled and sinned, so should justly be punished (in hell). But God had mercy and sent his only son to voluntarily die for us and so wipe away our sin and save us. If you have any conscious you will accept his deep sacrifice and worship him (or else his sacrifice won't count and he will be forced to let you go to hell). There is certainly a logic to this, as well as inducements to feeling guilty. Christianity holds so long as one remains convinced by this logic. Looking at it now this logic seems faulty on many levels, but nevertheless it was strong enough to hold me for many years.
There is no proof for logics like these, but external proof is too much to expect for any logic. So why do some people hold onto these logics so strongly while others reject them? Why are some logics that used to be plausible (and believed to be fact), such as Greek myths and Odysseus's adventures, no longer seem believable? Christianity has held sway for a good 2000 years. It certainly changed and mutated a lot initially, and continues to evolve. These changes are likely driven by what logic people find plausible. We drop or modify parts we no longer find plausible, like a god called Yahweh, who was one of the gods, fighting for his chosen people, and make him now into the one and only God. Other parts we sort of brush aside, like Jesus spending a lot of his time casting out demons which apparently were a regular part of society (like witches). We add a trinity that is really one and three at the same time, even though we can't say exactly what the one and the three are. It is a logic that has certainly swayed great minds.
Longevity tends to engender the assumption that it will go on forever. But I wonder if the days of Christianity are numbered? Is its logic is starting to fray in people's minds? Are people seeing the holes and starting to categorize it as one of the great myths of our time?