This is from a Agnostic Atheist's view point.
So I'm going to start this off by getting to the question, how do you know any of the Pagan branches and gods are real? Usually with other religions like Christianity or Islam it is because of their holy books, but from what I see with Pagan branches like Asatru/Heathenry for example people join it because of their Heritage, but I don't understand how that could possibly be evidence of the existence of gods like Odin or Zeus. Others I seen enter these branches because of what they call spiritual connections, stuff I use to believe in before became Agnostic Atheist and turned away "spiritual connections" for scientific facts and evidence. I'm sorry, I just don't understand this sort of thing anymore.
Some might ask me how I don't understand this sort of thing if I'm Agnostic Atheist instead of just a Atheist, that's because I don't believe there is a god or any god, and any spiritual belief that involves our lives and this world is just stupid and we can understand our lives and this world using science and evidence instead of pure religious faith, but the thing is while I believe that most likely nothing happens after death and it's just nothingness and blackness, I still think a afterlife is a possibility.
The reason I ask is because I listen to a lot of Pagan Folk Music, like Faun Wardruna and Forndom. To me it's a music thing, but I see so many people take it as a sort of spiritual path and faith.
Different Pagans will have different approaches since, as a rule, Paganism emphasises coming to your own conclusions. I'll share my own approach to things, but keep in mind that my views aren't representative of all Pagans.
My view on gods can be a little bit complicated, but to put it simply, I view gods as aspects of nature. To me, Erebus literally is darkness for example. So how do I know Erebus is real? Simple, whenever I'm in darkness, I'm surrounded by Erebus. At that point, asking for evidence or proof of these gods becomes something of an odd question. Look around you and you're looking at the gods.
Perhaps a more pressing question would be why I view these things as gods. There isn't a single answer to that. Firstly, I suspect that some combination of my nature and background makes me inclined towards theism. If I'm right, then I was always going to view
something as a god to begin with.
Secondly, I genuinely feel that there are elements of nature that are absolutely worthy of being viewed as gods. They inspire awe, fear and delight. They can be nurturing or destructive. When people say "but it's just thunder/darkness/the sea, etc" there's a part of me that feels they lack proper appreciation for those things. I don't begrudge them their views, I just view these things as "more" than they seem to.
Finally, I've always been drawn to folklore, mythology and story-telling. I feel that by honouring my gods, I'm keeping some of those folkloric things alive in my own little way.
Hope this helps