Succumb is suc- + cumb. Suc- is a form of the common prefix sub-, while sub- is derived from the Latin verb cubāre, meaning "to sit or lie down". Thus you succumb to your enemy when he has worn you down to the point where you are forced to lie down before him.
This root turns up in some other words too. To incubate something is literally to sit on it. An incumbent is a person currently sitting in the metaphorical chair of office. Concubine comes straight from Latin concumbere, "to lie together".
I felt sure that encumber would also belong to this family, but according to the Online Etymology Dictionary, encumber derives from Latin combrus, "barricade or obstacle".