The phrase, rest easy, is a command that means “relax, don’t worry, everything will be okay.”
As you know, rest means relax or take a break. Easy can mean “without anxiety” in several English phrases:
Rest easy, knowing that your money is safe with us.
Don’t be hard on yourself; take it easy.
Or like the famous song by Lionel Richie—Easy (Like Sunday Morning).
Rest easy is most often said when someone is anxious or worried—it is a reassurance.