Fathom was originally a land measuring term derived from the Anglo-Saxon word "faetm" meaning to embrace. In those days, most measurements were based on average size of parts of the body, such as the hand (horses are still measured this way) or the foot (that's why 12 inches are so named).
A fathom is the average distance from fingertip to fingertip of the outstretched arms of a man -- about six feet. Since a man stretches out his arms to embrace his sweetheart, Britain's Parliament declared that distance be called a "fathom" and it be a unit of measure. A fathom remains six feet. The word was also used to describe taking the measure or "to fathom" something. Today, of course, when one is trying to figure something out, they are trying to "fathom" it.
The ten and one hundred fathom curves on nautical charts have extra significance in that a ship's Restricted Maneuvering Doctrine usually requires notification of the Commanding Officer and Navigator when you cross them. Since they indicate that the ship is leaving deep water, it is usually the time to set extra navigation watchstanders to help ensure the ship stays out of the dirt.