The above commentary of Theophylact of Ochrid is beautiful and succinct, I just comment upon it:
to "fulfill" can mean 2 things:
a) to do carry out everything without omission; that is to say, to do all the 10 commandments meticulously, without dropping a jot from them. Imagine, for instance, in judo, or any martial art, you are given a rule to master some established set - say, 10 - of wrestling tricks, and you fulfill them, one after another, adding nothing and omitting nothing, so that at the end of your training course you become an complete judo wrestler, one of a centuries long ennobled tradition of the sport, with no innovation whatsoever.
b) to bring something to a completion/perfection, that is to say, see in something a latent potential for its making perfect and thus change it in order to make it perfect; but this change will not be a ruin of the mentioned "something", but on the contrary, a fulfillment of it, as, for instance, a ballet dancer did not destroy human movements, but perfected human movements to the level of being able to express more complex emotions and thoughts through them.
When Jesus says "I came to complete the Law", He does not mean that He came to fulfill everything of the Law in the first sense, for then He would have given a model of a complete Old Testamental virtuous man. However, this is no more relevant for the infinity of the divine Grace that came to humanity through Him. Therefore, Jesus says "I came to complete the Law" in the second sense, that is to say, seeing in the precepts of the Law a novel interpretation that would make man not only a good man in the Old Testament sense, but give him the ambition and authority of becoming a son of God (John 1:12); paved a way for man to aspire becoming perfect as God Himself (Matt. 5:48). Thus, "do not commit adultery" in the old interpretation meant that husband or wife does not go with other women or men respectively; however, in new, fulfilled or perfected interpretation of Jesus the one who looks with a lustful eye to a man or woman has already committed adultery; thus, the Law became much more difficult, refined and demanding. The same with, "you shall not kill" commandment: before you were an observant of it if you had actually not committed a murder, but now, in Jesus' perfected interpretation, to hate somebody is to commit a murder - thus Jesus does not treat the actions themselves, but the inner source of those actions, the very depths of human heart, and by giving those impossible commandments He also gives His divine aid, His Grace that we may be able to fulfill them together with Him, for "impossible for men is possible for God" (Luke 18:27).
Thus, the previous commandments, or to say more precisely, the precepts of the Law of Moses in their previous, Old Testamental interpretation could be performed simply by human efforts alone, yet without inner transformation and transfiguration of the inner core of human heart, whereas, on the contrary, the new commandments of Christ, or to say more precisely, the precepts of the same Law of Moses in His novel, perfected interpretation can by no means be fulfilled by human efforts alone, but necessarily entail Christ's co-action in and with us, that together with His divine aid we may aspire of becoming as perfect as Him and our Heavenly Father.
There are many gray, innocuous stones at the seashore, but when the sea waves make them wet and the sun rays shine upon them, the stones reveal their hidden latent beauty and shine in many different colors - so also Jesus, by pouring the Holy Spirit upon the gray stones of Moses' commandments, made them shine with their perfect latent beauty.