Jesus is saying, in plain language, that His teaching does not contradict the Old Covenant law, but it is the ultimate fulfillment of its spiritual intent. Even in the smallest matter, the smallest statement—the jot and the tittle—the law must be fulfilled.
Notice where His statement appears. Matthew places it immediately after Jesus' exhortation, "Let your light so shine before men, that they may see your good works" (Matthew 5:16). What if our works are good? Are we supposed to hide them? Then comes His statement regarding law. Is there a connection between good works and keeping the law? One would have to be quite obstinate to believe there is no connection between them. It is obvious that He is connecting good works with lawkeeping.
To strengthen the argument, He mentions righteousness in verse 20. What is the Bible's definition of righteousness? Psalm 119:172: "All Your commandments are righteousness." Thus, sandwiched between righteousness and letting one's light shine comes an explanation that He did not come to do away with the law but to fill it to the full, to help us understand its ultimate application—its spiritual intent.
Is it possible to keep the law in its spirit without also keeping it in the letter? It cannot be done. One must first keep it in the letter before learning how to keep its spirit.