The whole point of cryptocurrency is decentralized ownership. That’s the big breakthrough in technology, it’s the whole point of it, I can try to ELI5 how that works if you want to, but for the moment I’m just going to assume you accept that cryptocurrency can demonstrate ownership.
NFTs are an extension of that, except they can’t be split or traded by one another, i.e. they’re non-fungible. Therefore you can by definition prove ownership of those tokens, as that’s the whole point of the technology, which again, if you’re curious I can try to explain how it works.
How does ownership of those tokens transfers to ownership of something else? Well, that’s an excellent question, and the answer is that it happens in the same way that a piece of paper grants ownership of a house. There’s no innovative technology behind that piece of paper, but still everyone would agree that it grants ownership, and the reason is that the authority that enforces that chose to respect that piece of paper. Nowadays this is mostly databases and the piece of paper is just generated from the records there, but this is very insecure as anyone with database write access (or access to the physical folder containing the documents in case of old paper deeds) can transfer ownership. NFTs solve this because only the owner of a token can transfer it to someone else, so they’re inherently safer than any of the alternatives.
Again, the technology is great and has millions of excellent applications, but people use it for pyramid schemes and scamming others, but people do that with any piece of technology.