"Stay blessed" signifies "May you stay in God's favor."
In older English, it was worded as "May God never turn his face from you." Stay blessed is more imperative when a speaker wants to express to the person he is speaking to – Is doing good (behavior).
Stay blessed is usually addressed with religious beliefs in mind; it can also be a generic phrase. If you say this phrase to someone, you denote that you would like them to continue having good things in their life.