“Do therefore this that we say to you. We have four men who have a vow on them. These take, and purify yourself with them, and be at charges for them, that they may shave their heads, and all will know that there is no truth in the things of which they have been informed concerning you, but that you yourself also walk in an orderly way, keeping the law.”
So their suggestion was that he meet the costs of four young Jewish Christian men who were involved in a Nazirite vow. This would involve him purifying himself in the Temple for seven days with them for only then could his offerings be acceptable. And he would thus be sharing in their last week of consecration before they shaved their heads, and presented the hair to God with appropriate sacrifices. It would be a sharing in their consecration but not a strict participation. He would not be taking a Nazirite vow. Yet he would be offering sacrifices and thankofferings and rededicating himself and expressing oneness with these young men.
Bearing the costs of young Nazirites was a recognised form of showing generosity and giving to God among the Jews. King Agrippa I had used this method in order to make himself popular with the Jews. It was a regular practise among the more wealthy Jews who wanted to express their gratitude to God, and especially with those who wanted to be seen as pious. And it was a true kindness, for the offerings that had to be made by a Nazirite could be costly, and many had entered into their dedication in the hope that some noble benefactor would come forward at the end to meet their costs. No one would think it strange then if Paul did so, or consider that Paul was trying to muscle in on the dedication of the young men. All would see it as a good and noble and fully Jewish action.
And the result would be that all Jewish Christians would recognise that Paul was truly faithful to, and approved of, the customs of the Jews with regard to the Law of Moses. They would have their doubts laid aside.
Someone might cavil at the thought of Paul offering sacrifices. But we have reason to believe that he had observed the Passover at Philippi (Acts 20:6). And we must remember that the One Who certainly had no need to do so, regularly did participate in sacrifices, as we know for certain from the Last Supper. He did it in order to fulfil all righteousness, just as He was baptised for the same reason (Matthew 3:15). The full revelation of the end of all sacrifices was a truth which had not yet burst on the church. And we can be sure that all Jewish Christians within range of Jerusalem constantly offered sacrifices as worship and dedicatory offerings, and that the Apostles, including Paul, approved.