The river is famous to the fish. (1)
Why aren't the fish famous to the river? Can this fame go both ways? Can the river be famous to the fish at the same time they take it for granted?
The loud voice is famous to silence, which knew it would inherit the earth before anybody said so. (2-4)
The loud voice must belong to people, right? No one else speaks with a loud voice except us. Really, when's last time you had to ask the squirrel in the seat behind you to pipe down? Yep. Never. See what we mean? This line seems to say that silence has a deeper (maybe even better?) sense of time and fate than people do. So maybe Nature—and here we're talking about that image of nature as being peaceful and without humans in it—is described as knowing more than humanity.
The cat sleeping on the fence is famous to the birds watching him from the birdhouse. (5-6)
Cats sleep. That is just what they do. If you've ever had a cat for a pet, you know this. They also eat birds; it's just the circle of life, folks. In this case, maybe fame does go both ways—up and down the food chain. The cat is famous to the birds, because it's a threat. But the birds just might be famous to the cat because they're tasty.