My mother took care of the dog while I was on the trip.
Literal
Trip [possessive-の] during, mother [subject-が] dog [possessive-の] care [object-を] did [for-me-てくれた].
旅行の間 (during the trip) uses 間 for a duration. 犬の面倒を見てくれた combines 面倒を見る (to take care of) with てくれた (did it as a favor). The speaker's gratitude for their mother's help is built into the grammar.