To fathom something is to understand it thoroughly. It's usually used in the negative, as in "I can't fathom why he doesn't want to go along with us."
Fathom is from Middle English fadme, from Old English fæthm, "outstretched arms." The Old English and modern English noun also refers to a unit of length used to measure the depth of water. The sense of intellectual depth comes from that meaning — to understand or fathom something thoroughly is "to get to the bottom of it."