The sound of a jet taking off gets on my nerves.

Literal

Take-off do jet-plane [possessive-の] sound [subject-が] my nerves [to-に] touches.

神経にさわる (to get on one's nerves, literally 'to touch the nerves') is a vivid expression for irritation. 離陸するジェット機の音 is a noun phrase with a relative clause modifying 音.