The Law Fulfilled!

What did Jesus mean when he said “...I came not to destroy but to fulfil.”? If he did not destroy the Law or the prophets, then certain components like the Sabbath-day requirements and celebrations of the Feasts must still be binding and operative together with numerous elements of the Mosaic Law! However, such a view would also contradict everything that we understand from the balance of the New Testament.

As disciples of Jesus Christ under the new covenant, how then are we to respond to being under grace and law?