Leave the care of the kittens to me while you're away.
Literal
Absence-during kittens [possessive-の] care [topic-は] leave-to-me please.
任せてください (leave it to me / trust me with it) is a confident offer of help. 留守中 (while absent) uses 中 as a time suffix. 子猫 (kittens) is a compound: 子 (child/young) + 猫 (cat).