Unclean has the meaning of being dirty, or contaminated, having come in contact with the blood of an animal or man. The animal sacrifice was a substitute sacrifice for those sins of the people until Christ's sacrifice eliminated it.
The sin offering was necessary, but required cleansing of the "blood", the death of the animal, and the sin of the people. Ultimately sin is death. The priest could come into the camp after being purified (washed), but could not enter into the temple, into the presence of God before the time required to be completely clean of that contact with sin.
So, being unclean until the evening meant that the priest could enter into the temple at evening, but could not perform his duties in the temple until then.
Notice the proscribed time necessary for being completely cleansed of contact with a dead man in verse 11. Num. 19:11-12,
" 11 He that toucheth the dead body of any man shall be unclean seven days.
12 He shall purify himself with it on the third day, and on the seventh day he shall be clean: but if he purify not himself the third day, then the seventh day he shall not be clean."
Purifying required the washing of water, and was looking forward to the purification of water baptism under the new covenant of the gospel of Christ.