채우다 is filling a container, a stomach, a schedule or a role, and -게 gives the manner. Most of these adverbs mean 'amply': 풍성하게 채워진 (richly stocked), 배 두둑하게 채우고 (stuffed full), 든든하게 채워주시러 (coming to fill the spot solidly). It is used just as readily for abstract filling — 한국어로 이렇게 채워드릴 수 있었구나 is about filling a slot in Korean.
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