IPA: /ˈwɒtʃfəli/
KK: /ˈwɑtʃfəlɪ/
In a way that shows careful attention and alertness to what is happening.
The teacher watched the students watchfully during the exam.
Watchfully → It is formed from "watch" (from Old English *wæccan*, meaning to be awake or to guard) and "-fully" (meaning in a manner full of). The word "watchfully" means in a manner full of vigilance or attentiveness.
Think of being 'awake' and 'guarding' something — that's why watchfully means to be attentive and vigilant.