Paul alludes to Leviticus 19:18: the Law is fulfilled in one commandment, “Love your Neighbor As Yourself” (Gal 5:14). To fulfill the Law is not to “sum it up” but rather to complete (Note1). The verb “fulfill” (πληρόω) in verse 14 is a perfect passive, indicating that the completion of the Law has already been accomplished when believer “loves his neighbor as yourself.” Paul’s point is that if one is loving one’s neighbor as themselves, then they are already doing the “spirit of the Law.” By walking in the Spirit the believer is already fulfilling the whole law, but the requirements of the Law were never required for the Gentile believer in Christ (Witherington, Galatians, 381).
There was a lively debate in the first century on how to sum up the law, Jesus deals with this when he is asked what the greatest commandment is (Matt 22:34-40). There Jesus says that the Law and prophets “hang” on these two commandments, perhaps “hinge upon” is another way of translating the verb.
Defining who is one’s neighbor was also a point of discussion. An example of this is the context of the Good Samaritan parable. Jesus is asked by an expert in the Law to define “neighbor.” The lawyer likely understood the word to refer to fellow Jews, since that is what it means in Leviticus. Jesus expands this to include anyone who is in need. It is possible Paul has fellow-Christians in mind here, given the context of factions within the church (5:15, 26), but he will expand the doing of good in 6:10 to everyone, but especially the “household of faith.”
This verse is the most quoted verse from the Pentateuch in the New Testament, despite the fact that it is almost never referred to in contemporary Jewish texts. Perhaps this is because Jesus himself stressed love of neighbor as a fulfillment of the law! Paul’s point is not, “if you want to keep the law, love your neighbor.” He has said repeatedly that the age of the Law is done and over with and the one who is in Christ is free from the Mosaic Law.
My guess is that most contemporary Christians think that by loving their neighbor they are somehow keeping the Law of Christ. What motivates “loving your neighbor” seems to be the main point here – why do you serve others? If it is to keep a “Law” or because it is “expected of you,” then it is possible you have missed the point altogether.
Note 1: A number of manuscripts (including D F G and the Majority text) have πληροφορέω, often used with the sense of fulfilling prophecy. The 12th century mss 365 has ἀνακεφαλαιόω , to “unify” or “sum up.” The UBS text is more likely, with support from P46 and A B C.