In Buddhism and Hinduism it is commonly said to experiment and see if what the masters teach is true or not. This is basically what I find myself doing. I practice concentration meditation from the Buddhist tradition, and it's well laid out what is supposed to happen....
First, concentration meditation will bring you a lot of clarity and mental tranquility. This part I can verify as true. There are lots of unexpected benefits that happen. I can for sure verify that it is very good for your mind to meditate like this.
The next step is to leave behind your worldly pleasures/desires/addictions, and supposedly you will find a happiness/joy so great that all the things you clung to before could never compare. I haven't done this yet. I've tried, I've probed it, and I can definitely say that abstinence from things does seem to make me feel happier. But I haven't let go all the way, so this part I cannot confirm nor deny. It seems very probable though, and I see enough anecdotal evidence from Monks and other religious individuals to have good faith that this is true.
Now for the more unbelievable parts...
It's said that with devoted concentration meditation, you can attain psychic powers. Astral projection being one. Others being clairvoyance and other strange stuff. There are people right here on reddit that claim to have experienced these things. I have not. However I also have not been meditating seriously for that long either. Both Buddhism and Hinduism acknowledge these things, and what is sort of interesting is that they both say "pay no mind to them, they are distractions on the path".
And that brings us to the ultimate goals of these practices, which is nirvana or moksha. In Buddhism, the wording is of realizing "emptiness" and "no-self". In Hinduism, it's talked about more along the lines of being "God-conciousness". Is there such a thing? I don't know. Many people describe similar stuff in trip reports (which is why I read this sub). So maybe? I can't yet say though.
So what to make of these claims? I have no idea. What I do know so far is that meditation has been very beneficial so far. Tranquility of mind, greatly increased focus and attention, and instructions to "love all the people around me". I can verify for you, that these preliminary things are pretty legit. All the other stuff... I don't know.
Hope that helps.