This is the 164th most frequent Korean word.
더욱
Even more; increasingly.
In this sentence, '더욱' is used as an adverb to intensify the degree of the adjective '추워졌어요' (gotten colder), suggesting the degree has increased.
날씨가 더욱 추워졌어요.
The weather has gotten even colder.
In this instance, '더욱' modifies the possibility of the action '성장할 수 있습니다' (can grow), indicating that the extent of growth will be larger.
노력하는 사람은 더욱 성장할 수 있습니다.
People who make an effort can grow even more.
Here, '더욱' enhances the verb phrase '좋아질 거예요' (will become better), emphasizing the improvement compared to the current state.