For anyone curious about how this phenomenon works:
The "Finger of Death" refers to a natural occurrence called a brinicle, an underwater icicle formed when supercooled saline water is expelled from sea ice. As this denser, extremely cold brine sinks, it freezes the seawater around it, creating a tube-like structure that extends downward. Anything in its path, like sea life on the seafloor, can freeze as the brinicle advances.
The formation speed can vary based on factors like temperature, salinity, and water depth. Typically, a brinicle grows at a rate of a few centimeters to around 10 centimeters (4 inches) per minute, though local conditions in Antarctica can affect its exact speed.
EDIT: I copied this explanation from the net.
How do they get the sound effects of the finger growing?
Maybe it’s the actual sound of the water freezing, recorded with a highly sensitive microphone and then amplified through editing? It could also be possible that the editors used a different sound and layered it onto this particular clip or scene.