This is the 237th most frequent Arabic word.
قائلا
"Qāʾilan" (قائلا) means "saying" or "speaking."
In this sentence, 'قائلاً' describes the concurrent speaking as the person is looking at her, detailing the action to make the scene vivid.
نظر إليها قائلاً: 'أنتِ محقة'.
He looked at her and said, 'You are right.'
The word 'قائلاً' links the physical action of waving to the spoken words, indicating simultaneous actions.
ولوحَ بيده قائلاً: 'إلى اللقاء'.
He waved his hand saying, 'Goodbye.'
Here, 'قائلاً' is used to emphasize the manner of the speaker's action (saying), effectively acting as a gerund meaning 'saying'.