When a person is at a loss for words, that person is speechless.
Speechless does not mean unable to speak (i.e. a person who is mute); it means someone who is too overwhelmed to speak, or at the very least, unable to say anything profound enough to suit (fit) the occasion.
Example 1: Margaret: "I saw a car accident where five people died. It was horrible...!" Peter: "I'm at a loss for words. That's terrible, I'm so sorry to hear that."
Example 2: Patricia: "My friend in Boston won a million dollars in the lottery!" Donald: "I'm at a loss for words... that's incredible!"