“See that you say nothing to anyone.” Jesus often said this when he healed people. (cp. Matt. 9:30, blind men; Mark 7:36, a deaf person; Luke 8:56, a dead girl). On the other hand, he told some of the people he healed to spread the news (cp. Mark 5:19). The reason Jesus told people to not talk about their healing is never explicitly stated, and there are likely various reasons for it. One reason would be the often infectious doubt and unbelief that comes from scoffers who hear of the deliverance which could adversely affect the person who was healed. Another reason would be the personal privacy of the individual, who would often be immediately elevated to “movie star status” in their community, as happened to Lazarus. Still another reason would be likely especially in cases such as this healing, the need for Jesus to not be thought of as “unclean” or worse, as “contagious,” in his culture. Touching a leper made the person unclean (Lev. 15:7), and since the cause of leprosy was unknown, not only were people with leprosy scrupulously avoided, but no doubt sometimes people who touched lepers were looked upon with suspicion and avoided.
“as a testimony to them.” Jesus’ healing the leper would be one more testimony that he was the Messiah. However, the Greek wording can, and often is, understood in a negative sense, “as a testimony against them” (see Robert Guelich, Word Biblical Commentary: Mark). The priests were against Jesus, and their not seeing the miracles that he did as proof of who he claimed to be was against them.