Lake Leman (a.k.a. Lake Geneva) is right outside the walls of the castle where the speaker and his brothers are imprisoned.
He says that the water of the lake is a thousand feet deep right outside the prison walls – at least, that's as far as the "fathom-line" was able to measure.
The white "battlement," or castle wall "inthrals" or encircles the waves of the lake.
Between the dungeon wall and the deep lake, the prisoners are in a "double dungeon." It's a little like Alcatraz used to be – if you escape over the wall, you still have a deep body of water to deal with.
The speaker says that their dungeon was below the surface of the lake (although, in actual fact, the dungeon of Chillon Castle isn't below the water level – but Byron's allowed to take liberty with historical fact).
He compares their dungeon to a "living grave" – being imprisoned there is like being buried alive.
Being so close to the lake, they can hear the sounds of the water over and around them in their dungeon.
The speaker occasionally feels spray from the lake through the grate in their dungeon when it's windy outside.
When the weather is really bad and he can feel the castle rocking in the bad weather. He says that he's not at all afraid, because he'd welcome death by any means – even having the dungeon cave in on his head during a terrible storm.