When you clench something, you grip, press, or squeeze it. You might clench your fists when you're angry or clench your teeth when you're worried, for example.
People often clench parts of their own bodies, but they can also clench other things. Your grandmother might tend to clench you in her arms when you visit her, and a toddler might clench his stuffed animal to his chest when he falls asleep at night. The Old English root of clench is beclencan, which means "to hold fast or make cling," and it has the same origins as the word cling.