1. To have a calming effect - "Oh, well," said the captain, soothing his children, "We'll get ashore all right."
The verb soothe means to relieve, to gently calm. The captain's words intend to have a calming effect on his children, to make them feel better.
2. To get up or raise oneself - The captain, rearing cautiously in the bow, after the dinghy soared on a great swell, said that he had seen the lighthouse at Mosquito Inlet.
One of the meanings of the verb "To rear" is to raise upright, to get up.
3. To search or peer at thoroughly to locate something - But at last there came a wave more gentle than the others, and when at the crest of it he swiftly scoured the western horizon.
To scour is to do a thorough search in a place, text, etc., usually quickly, in order to locate something.
4. To crouch or sit with bent knees - The cook squatted in the bottom and looked with both eyes at the six inches of gunwale which separated him from the ocean.
To squat is to crouch or sit with one's knees bent and one's heels close to or touching one's buttocks or the back of one's thighs.