This r/asklinguistics post answers my question.
gnorrn. 4 points 6 days ago
De Vaan, Etymological Dictionary of Latin, gives a derivation from Proto-Italic *kampo-, "field", and also supplies Balto-Slavic cognates meaning "corner".
This suggests that the semantic development into Proto-Italic may have been something like "bend" -> "corner" -> "area bounded by corners".
ecphrastic. 6 days ago
The source you've quoted lists campus 'field' as probably properly a 'surrounded expanse'. "Surrounded" seems to be the critical semantic similarity here. A corner is an area surrounded on multiple sides. A field is a type of landscape, which someone might define in opposition to the other landscapes it borders, such that you could imagine campus originally meaning a minimally vegetated area surrounded or bordered by thickly vegetated areas, and eventually shifting slightly to mean the terrain of the minimally vegetated area itself.
The English word clearing lies somewhere in that same spectrum of meaning. A clearing strictly speaking is a small treeless area in a forest, especially from logging. It's not too far-fetched to think it could come to mean something like 'meadow', but right now it's a word that defines treeless terrain in opposition to adjacent forest.