Here is a brief summary of my interpretation of this text. While I have already written a more extended article on it, I have here summarized arguably the most crucial points in 4 simple points.
1) Since the Greek word translated "fulfill" means that in virtually all of its usages in the Gospel of Matthew (Matt. 1:22; 2:15, 17, 23; 3:15; 4:14; 8:17; 12:17; 13:35, 48; 21:4; 23:32; 26:54, 56; 27:9), the burden of proof is on the reconstructionist to prove that it means something other than that in this passage.
2) In light of the context of vv. 17-20, its most likely significance is as a preface to the antitheses of vv. 21-48.
3) Since the sayings on divorce and remarriage (vv. 31-32) and the prohibition on all oath-taking (vv. 33-37) explicitly contradict Old Covenant norms (see especially Jesus' commentary on Deut. 24:1-4 in Matt. 19:1-12), it would be extremely strange for the preface of vv. 17-20 to emphatically affirm the ethical and preceptual continuity of the New Covenant with the Old, only for the body for which it served as a preface to explicitly contradict those norms.
4) The most likely meaning of v. 17 is therefore, not to emphatically affirm straightforward continuity except where otherwise specified, but to anticipate the objection that apparent abrogation of surface commands of the Old Covenant do not constitute a genuine abrogation, since the ultimate purpose of the Old Covenant was to point to Christ, who functions as its fulfillment (Gal. 3:21-23).