First, they were to set themselves apart by their behavior. Because they were presenting themselves before God and representing the people, it was important for them to be "clean." In our modern context, I don't think we have to worry about whether they've bathed or shaved. But the notion of having prepared themselves to approach God is important. Pastors, like these Levites, are fallible humans, so there is a sin offering. Jesus died for the sins of your pastor just as He died for your sins.
They were also set apart in the sense that they were not to perform other work and the Israelites were commanded to provide for them. Our modern culture has gone awry to both extremes with this. On the one hand we have pastors who are featured on "Lifestyles of the Rich and Famous" (OK, I just dated myself...it was a show in the 80's). On the other hand, we have pastors who must work a second or third job to provide the essentials for their family. Neither of these situations honors God.
They were also set apart to represent the firstborn, those who God spared in the final plague of the Exodus. Their lives were to be completely devoted to God -- just as completely devoted as if their lives had been burnt up on the altar. As Paul tells us in Romans 14:8, "If we live, we live for the Lord; if we die, we die for the Lord. Therefore, whether we live or die, we are the Lord’s." He reiterates the same concept in Philippians 1:20, "My confident hope is that I will in no way be ashamed but that with complete boldness, even now as always, Christ will be exalted in my body, whether I live or die." In this, those who serve the Lord are meant to be an example to all believers of a life consumed by God. A life more concerned with exalting Christ than with its own existence.