What did Jesus mean here? How did Christ fulfil the Law? What does that mean for the Law itself?

Christ Fulfilled the Law
The Law can mean: a) God’s decrees, b) Scripture, c) a Covenant, d) the Mosaic Covenant (or Covenant with Israel at Sinai), or e) God’s moral law. Given the context in the Sermon on the Mount it is almost certainly “d,” the Mosaic Law; yet with a twist.

To abolish means to unyoke, as in unyoking an animal from a cart. As such it means to break or destroy what was.

To fulfil means is to be full or to meet.

The Jewish leaders of Jesus’ day feared He was a religious revolutionary who would upset their cherished possession, or rather their misinterpretations and additions (Mt 23:4; Mk 7:7), for Jesus, being perfect, never broke God’s Law. He would be much more radical and still more conservative than they thought.

Christ fulfilled the Law by doing what Adam, doing what the descendants of Abraham, and of Israel and the Kings could not do—be that perfect covenant partner. No human can by their works “fulfil the Laws demands” (“Rock of Ages”).

Christ could fulfil the Law, as Matthew is keen to point out, because He was the lawgiver greater than Moses.

The New Covenant
In fulfilling, or meeting, the demands of previous covenants, Jesus inaugurated the promised New Covenant (Jer 31:31; Ezk 36; Heb 8 et al).Jesus’ life and ministry marked a watershed or transition period between the covenants (it was inter-covenantal). When He died the veil was torn. After He ascended the Spirit was given. There is a newness in the New Covenant. New (kainos) means something new in kind, like a new invention; it isn’t new (neos) as in a new type of car but a new form of travel like a teleporter.  (The NC doesn’t abolish, replace or succeed the Old, it fulfils the promises of the Law and Prophets. It is the direct continuation of God’s plans).

According to Gal 3:15–29 the Law of Moses was temporary and served the purpose of exposing our sin and making the promise to Abraham essential. It also has a guiding quality.

As such certain aspects of the Law of Moses were no longer necessary. Since Christ was the sacrifice for sin and the Holy Spirit now made believers the living temple of His presence the Temple was obsolete and hence the ceremonial system. The dietary laws (an external sign of holiness) were no longer necessary for Christ taught that holiness flowed from Christ’s imputation and through a new heart cleansed from within by the Spirit. (Scripture emphatically declares this in Mk 7:19b, “Thus He declared all foods clean,” c.f. Acts 10). Circumcision as the covenant sign gave way to baptism, the Passover to the Lord’s Supper (Lk 22:20), and so the list could go on.

A Law Remains (The Law of Christ or Moral Law)
How then could Jesus say in Mt 5:18, “For truly, I say to you, until heaven and earth pass away, not an iota, not a dot, will pass from the Law until all is accomplished.” Likewise, how could he commend the “scribes and Pharisees” pursuit of righteousness, or holiness, (v. 19), and state this legal righteousness was needed to enter the Kingdom? (Truly, Christ is our righteousness and the Spirit enables us to live righteously, thus guaranteeing our place in the Kingdom, both present and eternal).
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In saying that He would fulfil the Law of Moses and yet the law would never pass away Jesus is commending to us the Law of Christ, or the Moral Law (Gal 5:14, 6:2). While the Law of Moses as a whole has been fulfilled in Christ, a law remains which is the moral law found within it (see 2nd London Baptist Confession, ch. 19, for the classic Christian understanding of the threefold division of the Law). This is binding upon all believers to follow as our guide to holiness by the Spirit.

, …The Israel of God. (Galatians 6:16)

Who did Paul have in view in his benediction? After all the Galatians were comprised of both believing ethnic Jews and believing ethnic Gentiles.

The NLT says, “May God’s peace and mercy be upon all who live by this principle; they are the new people of God.”
The NIV says, “Peace and mercy to all who follow this rule—to the Israel of God.”
The ESV says, “And as for all who walk by this rule, peace and mercy be upon them, and [which can translate ‘even’] upon the Israel of God.”

It is clear from the context of Galatians that Paul is referring to the Galatian believers as “the Israel of God.” The term Israel can mean a number of things in the Bible: a definition (one who struggled with God), a man (Jacob), a nation, the Old Covenant community, a political entity or generally God’s people. Here it is in the last sense that Paul is using “Israel.”
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In fact, this is a great sub-theme of Galatians, especially, chs. 3–4. Contrary to the Judaizer’s false gospel, one doesn’t have to become Jewish (i.e. the Old Covenant community, which ethnic Israel had embodied) to be saved or be part of God’s family. God’s people are those who relate to God through faith in Christ/Gospel/NC. There is a newness in the NC. It is not merely a reforming of the OC. Christ had fulfilled the Old Covenant and ushered in the New Covenant governed by His 12 Apostles (think 12 tribes of Israel). We become Abraham’s offspring through faith (3:7) and there is no saving or meritorious distinction between “Jew and Gentile” (3:28), we are all “one in Christ Jesus.” Together we form God’s “new creation.” (Gal 6:15). Just as the word ‘church’ was used in the Greek Old Testament to speak of Israel (OC people of God) so today we can speak of Israel as referring to the Church (NC people God). From a NC perspective these are interchangeable words. Throughout Galatians Paul has been arguing that the Gospel produces a new multi-ethnic people of God who are justified by faith in the Messiah/Christ and live in accordance with the Law of Christ (moral law) by His Spirit. To enjoy the benefits of this Covenant one did not have to go backward in salvation history but forward. (Yet understandably transitions are not always the easiest to perceive when we are in the midst of them, Lk 5:3739, Acts 15).

This isn’t replacement or supersessionist theology but fulfilment and continuationist theology. Jesus was Jewish. The earliest New Covenant believers (until Acts 11:19) were Jewish. Though many Gentiles believed and joined Israel under the OC under the NC this became a fuller ingathering (c.f. Isa 49). Since Abraham/Moses there had always been a mixture of ethnic Jews and Gentiles in the OC community (Israel), because God’s plan of salvation had always been to redeem a people for Himself from every tribe and tongue and nation (Gen 12; classic view 1689.26.1). This he did progressively through Covenants, of which the New Covenant is most expansive.

What then of ethnic Israel? Paul addressed this in Romans 9–11 (Romans very much being an expansion of Galatians). You can see a brief visual depiction of this here. In short, the faithful remnant of ethnic Israel under the OC believed in the NC (the early Jewish believers, Acts 1–9). While some ethnic Jews presently believe many do not. Yet at the end of the age Paul envisions a great revival of ethnic Jews and their ingathering into the Body.