Things that sicken you make you feel upset or ill. It might sicken you to realize that you've hit a squirrel with your car.
When things sicken you, they either make you feel terrible — disgusted or horrified — or they make you physically sick. Eating a piece of moldy bread or some potato salad that's been sitting in the sun too long will probably sicken you. Reading a story about childhood hunger and poverty in the newspaper can also sicken you. The earliest meaning of sicken was "fall ill," from an Old English root, seoc, "ill, diseased, weak, or troubled."