키우다 is raising or growing something — a pet, a child, a plant. The -아서/어서 clause gives how it began: 데려와서 키우다 (bring the animal home and raise it, twice here), 심어서 키우다 (plant it and grow it), 통에 흙을 담아서 키웠는데 (grew it in a pot of soil). The same ending also supplies a reason, as in 고양이를 너무 좋아해서 키우게 된 거야? Contrast 크다, which is the thing growing by itself.