Fulfill the law does not only mean "comply with" the law, but includes the notion of completion. Of bringing something to its logical and ultimate conclusion.
Here are the meanings from BDAG:
to make full, fill (full)
to complete a period of time, fill (up), complete
to bring to completion that which was already begun, complete, finish
to bring to a designed end, fulfill a prophecy, an obligation, a promise, a law, a request, a purpose, a desire, a hope, a duty, a fate, a destiny, etc.
Thus until Christ, the law was missing something, and he completed it. What was the law missing? The Law is God's word, but Christ is the word made flesh, that is, Christ is also the torah made flesh, and in him, the law as written text is not only obeyed, but realized -- turned into a real, living embodiment.
In this way, we can see the law as a type of architectural building plan for a temple, and then Christ is that temple living in the world.
Or we can think of the law as a series of requirements and Christ is the perfect life that lives out these requirements.
Or we can think of the law as a prophecy and Christ fulfilling the prophecy. Thus the law was created for Christ to fulfill it - it is the set of standards that God requires. Published ahead of time so all of creation would recognize Christ as the only one who fully met these standards to his core.
This is why we have the hard sayings of the sermon on the mount, that lusting after a woman is the same as committing adultery with her (Matt. 5.27-28). E.g. that the law must be kept in the heart (Proverbs 3.1). E.g. the requirements of law are not superficial, but penetrate all the way to the heart. Obviously no one can meet these requirements except for Christ. Many can live a life without committing adultery in the flesh, but no one except Christ can keep the law also in his heart, to the very core of his being. Many can abstain from outwardly laboring on the Sabbath, but Christ is the only one who rested perfectly in his heart. Many can avoid killing, but Christ had no hate in his heart, etc. Thus Christ did not only meet the superficial form of the law as understood by man, but the true form of the law as seen by God, who looks at the heart (1 Sam 16.7).
So it is foolish for us to try to keep the law with our own superficial obedience, which is like filthy rags in comparison to Christ's obedience. Our hearts are deceitful (Jer 16.7) and so at best we can only meet the superficial outward requirements of the law, we cannot embody the law. Therefore our attempts to live out the law leave it unfulfilled.
For us, if we are honest and recognize that the law must penetrate to the heart as well as to the outward form, then we see in it our disobedience and thus our death. Thus to us, the law signifies our death, but for Christ, it is a testimony that he has met the standards prophecied in the law to his very core. Then we fulfill the law vicariously, by the death of the flesh which frees us from the law, and Christ's new life within us that fulfills it.
 Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 828). Chicago: University of Chicago Press.