My neighbors asked me to take care of their plants while they were away on vacation.
Literal
Neighbor [from-から] vacation [in-で] away while [object-を] plants [of-の] care [object-を] do [favor-くれる] [so that-ように] was-asked.
~てくれるように頼まれる is the passive of ~てくれるように頼む — 'was asked to (do something as a favor).' A long subordinate clause 休暇で留守の間 sets up the temporal frame. 植木 ('garden plants, potted plants') is a common noun in neighborhood-favor contexts in Japan.