He took the vow so people would know that he kept the law of Moses/God
Acts 21:24
He was forced to do that by a mob of angry Jews. They couldn't even wait until it was finished and drug Paul out to kill him.
Act 21:22 What is it therefore? the multitude must needs come together: for they will hear that thou art come.
Act 21:23
Do therefore this that we say to thee: We have four men which have a vow on them;
DO WHAT WE TELL YOU TO DO!
Act 21:24 Them take, and purify thyself with them, and be at charges with them, that they may shave their heads: and all may know that those things, whereof they were informed concerning thee, are nothing; but that thou thyself also walkest orderly, and keepest the law.
The leaders of this multitude of people tell Paul what they want him to do...and if you want to live, you do what a multitude of people tell you to do...this is a mob.
The Nazarite vow was not completed because Paul definitely didn't sacrifice any animals:
Num 6:9 And if any man die very suddenly by him, and he hath defiled the head of his consecration; then he shall shave his head in the day of his cleansing, on the seventh day shall he shave it.
Num 6:10
And on the eighth day he shall bring two turtles, or two young pigeons, to the priest, to the door of the tabernacle of the congregation:
Num 6:11
And the priest shall offer the one for a sin offering, and the other for a burnt offering, and make an atonement for him, for that he sinned by the dead, and shall hallow his head that same day.
Num 6:12 And he shall consecrate unto the LORD the days of his separation, and shall bring a lamb of the first year for a trespass offering: but the days that were before shall be lost, because his separation was defiled.
Num 6:13 And this is the law of the Nazarite, when the days of his separation are fulfilled: he shall be brought unto the door of the tabernacle of the congregation:
Num 6:14 And he shall offer his offering unto the LORD, one he lamb of the first year without blemish for a burnt offering, and one ewe lamb of the first year without blemish for a sin offering, and one ram without blemish for peace offerings,
Num 6:15 And a basket of unleavened bread, cakes of fine flour mingled with oil, and wafers of unleavened bread anointed with oil, and their meat offering, and their drink offerings.
Num 6:16 And the priest shall bring them before the LORD, and shall offer his sin offering, and his burnt offering:
Num 6:17 And he shall offer the ram for a sacrifice of peace offerings unto the LORD, with the basket of unleavened bread: the priest shall offer also his meat offering, and his drink offering.
Num 6:18 And the Nazarite shall shave the head of his separation at the door of the tabernacle of the congregation, and shall take the hair of the head of his separation, and put it in the fire which is under the sacrifice of the peace offerings.
As you can see, animals have no part of this until the 8th day.
Act 21:27
And when the seven days were almost ended, the Jews which were of Asia, when they saw him in the temple,
stirred up all the people, and laid hands on him,
Act 21:28 Crying out, Men of Israel, help: This is the man, that teacheth all men every where against the people, and the law, and this place: and further brought Greeks also into the temple, and hath polluted this holy place.
Act 21:29 (For they had seen before with him in the city Trophimus an Ephesian, whom they supposed that Paul had brought into the temple.)
Act 21:30 And all the city was moved, and the people ran together:
and they took Paul, and drew him out of the temple: and forthwith the doors were shut.
Yet still during the 7th day Paul was taken out of the Temple and did not return. Is the timing of this co-incidence? hmmmm....also note this mob of people are now violent to Paul proving they were to be feared all along!
Act 21:31
And as they went about to kill him, tidings came unto the chief captain of the band, that all Jerusalem was in an uproar.
Act 21:32 Who immediately took soldiers and centurions, and ran down unto them: and when they saw the chief captain and the soldiers, they left beating of Paul.
So they were beating Paul with the intention of killing him but the Roman's stopped this.