Looking at this in its context, we see that it is speaking first of the exodus of the Jews from their captivity in Babylon, but it is also a prophecy, speaking of a future exodus from a future captivity—from the Babylon that is now forming in this present age. This is very timely for us.
While the Jews were in their seventy years of captivity in ancient Babylon, they did not have the freedom or opportunity to maintain either ritual or spiritual cleanliness, as they would have had in their homeland. About 300 years later, the celebration of Hanukkah—meaning "dedication"—arose from the Jews' attempts to cleanse the worship of God following Antiochus Epiphanes and the Greek army's defilement of the Temple during warfare.
These verses are an urgent command, reminding them of their responsibility to cast off personal defilement of any paganism (or, as we would say today, any worldliness) picked up during their captivity. This had to be done to restore the true worship of the true God when they returned to Jerusalem.
The Living Bible translates verse 11 as follows:
Go now, leave your bonds and slavery. Put Babylon and all it represents far behind you—it is unclean to you. You are the holy people of the LORD; purify yourselves, all you who carry home the vessels of the LORD. (Emphasis ours.)
Their responsibility is clear. Who would be carrying the holy vessels? The priests. We need to note this inference since the whole church is a priesthood, and is so designated in I Peter.