Read the newspaper every day, or you'll fall behind the times.
Literal
Every-day newspaper [object-を] read [command-なさい], otherwise times [behind-に] fall-behind.
さもないと ('otherwise, or else') warns of consequences. 時勢におくれる ('fall behind the times') uses 時勢 ('current state of affairs'), more literary than 時代. ~なさい is the mild imperative.