。
I'll look after your dog until tomorrow.
Literal
Tomorrow [until-まで] dog [object-を] [humble-お]keep-預かりし [let's-ましょう].
お預かりしましょう uses the humble お + masu-stem + する pattern on 預かる ('to keep, look after'). ~ましょう here is an offer ('shall I / let me') rather than a group invitation. まで (until) marks the duration, distinct from までに (by, deadline).