Genius has a cool feature which lets users interpret song meanings cooperatively. It also allows you annotate the meaning of smaller blocks of the lyrics.
As it happens, Aerials has been interpreted quite extensively:
This is the main interpretation of the song:
The overall theme of the song is our mistaken views of ourselves as being individual and isolated from the rest – mankind and other lifeforms – and how we actually are a tiny part of something greater, from wich we come when we are born, and to wich we go when we die. It speaks of clinging to life and not accepting death, cause we see death as an ending, not as another step of existence. When we die, our matter is decomposed and is transfered to things like worms that feed on us, to the atmosphere in the form of gas, and to the ground and plants that feed on him. We are always moving through the cicle of existence, there is no stillness, only movement and transformation.
And as an example, these are the opening lines:
Life is a waterfall
We're one in the river
And one again after the fall
A user posted this as an interpretation:
The lyrics rush you with several meanings at once.
Life is a waterfall, what do we notice about waterfalls?
They fall downwards, so life brings us down. Isn’t it true? The actual reality of the things that go on in the world, murders, rapes, death, it can really bring you down if you acknowledge them. Another thing about a waterfall is that it never exists on its own, it always has a source, like a river or a stream. So if a waterfall is the acknowledgement of the terrible things in the world and letting it bring you down, what could the river be? The time before you realize what goes on in the world, often times this is your childhood. So when we’re one in the river, and we’re one again after the fall, it shows we are ourselves before and after the reality if life hits us and brings us down. Is that true for everyone? No, but think about it. When we were living as children with not a worry in the world, we were ourselves. Then once we began figuring out how the world worked and what happens in the world, we probably had a time of inner conflict where we had to figure things out, and after we figured/figure it out can adjust to it and find ourselves again.
An interpretation that another user commented on:
Actually, he says in the first line that life is a waterfall, meaning that life is the fall. Before life, we are all part of the calm river, then we are born and live through our life the way a waterfall falls to it’s inevitable end, which reunites everyone in death as we were before life. One with everything.
I guess the main thing here is that there isn't always a wrong or right when interpreting lyrics. However, Genius has a voting system similar to Stack Exchange, so I recommend you to go there and find out what the lyrics mean to you.