It means exactly what it says. He traveled over hills to get there and then under the mountain in the tunnels. He also lives under a hill in Bag End.
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According to the book The Hobbit by J.R.R. Tolkien:Smaug asks, "Who are you and where do you come from, may I ask?"Bilbo answered, "You may indeed! I come from under the hill, and under the hills and over the hills my paths led. And through the air, I am he that walks unseen. I am the clue-finder, the web-cutter, the stinging fly. I was chosen for the lucky number. I am he that buries his friends alive and drowns them and draws them alive again from the water. I came from the end of a bag, but no bag went over me. I am the friend of bears and the guest of eagles. I am Ringwinner and Luckwearer; and I am Barrel-rider."
The Hobbit is set about 60 years before most of the events in The Lord of the Rings.The Hobbit was published in 1937. The Lord of the Rings did not come out for over 20 years after that. It took Tolkien a long time to write the follow up.
That would be the men of Laketown and the Army of the Elven King. They have taken over the area to prevent the dwarves from getting out to get food. It also prevents Dain's men from joining the group under the mountain.
Not sure, but it does sound alot like him!
If you sang it a few times, or more than a few, it should stick. Your interest level would be the key.