The eye radical on the left is exactly what is doing the action: glaring, staring, eyes flying open in surprise. It groups 瞪 with 看, 瞥, 睛 and the whole family of vision and gaze characters.
Provides the sound, with a small tonal shift from dēng to dèng. The climbing meaning is incidental, though one can picture eyes climbing upward in shock — a fitting hook for the wide-open glare that 瞪 captures.
