The verb exclaim is from the Latin word exclamare, which means "to cry out." The English meaning is similar, to cry out, but with the added element of a strong emotion such as fear, joy, surprise.
In writing, when someone exclaims something, the punctuation mark used to express this vehement outcry is usually the exclamation point: "Stop!" you exclaim when your toddler climbs precariously on a chair stacked with books. "Not in this lifetime!" you cry out when your teenager wants to borrow the new car. You will exclaim "I'm so proud!" when your child graduates college.