Please keep me informed of everything that happens while I'm away.
Literal
Absence-during [possessive-の] happenings [topic-は] all me [to-に] inform [preparatory-ておいて] please.
知らせておいてください chains 知らせる (to inform) + ておく (do in advance/keep done) + ください — 'please keep me informed on an ongoing basis.' 留守中の出来事 (events during absence) uses 出来事 (events/happenings).