What almost every Christian fails to realize is that fulfillment of the Law - by us - is an essential component of the New Covenant. Take a look at the promise of the New Covenant:
Jeremiah 31:31-34 “The days are coming,” declares the Lord,
“when I will make a new covenant
with the people of Israel
and with the people of Judah.
32
It will not be like the covenant
I made with their ancestors
when I took them by the hand
to lead them out of Egypt,
because they broke my covenant,
though I was a husband to[d] them,[e]”
declares the Lord.
33
“This is the covenant I will make with the people of Israel
after that time,” declares the Lord.
“I will put my law in their minds
and write it on their hearts.
I will be their God,
and they will be my people.
34
No longer will they teach their neighbor,
or say to one another, ‘Know the Lord,’
because they will all know me,
from the least of them to the greatest,”
declares the Lord.
“For I will forgive their wickedness
and will remember their sins no more.”
So under the New Covenant, God's Law is written directly on our minds and hearts (vs the Mosaic Covenant, where God wrote on external stone tablets).
In addition to this, Christ is quite explicit that the Law isn't going anywhere - and condemns anyone who teaches otherwise:
Matthew 5:17-20 “Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. 18 For truly I tell you, until heaven and earth disappear, not the smallest letter, not the least stroke of a pen, will by any means disappear from the Law until everything is accomplished. 19 Therefore anyone who sets aside one of the least of these commands and teaches others accordingly will be called least in the kingdom of heaven, but whoever practices and teaches these commands will be called great in the kingdom of heaven. 20 For I tell you that unless your righteousness surpasses that of the Pharisees and the teachers of the law, you will certainly not enter the kingdom of heaven.
So God's Law serves as the very foundation of the New Covenant, and Christ was quite clear that nothing in the Law was going away. Indeed, in the Great Commission Christ tells us to go and teach others to obey the commandments:
Matthew 28:18-20 Then Jesus came to them and said, “All authority in heaven and on earth has been given to me. 19 Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age.”
Even Paul, who people fail to understand and constantly misquote, says that he is under God's Law.
Romans 7:22 For in my inner being I delight in God’s law;
Romans 7:25 Thanks be to God, who delivers me through Jesus Christ our Lord! So then, I myself in my mind am a slave to God’s law, but in my sinful nature a slave to the law of sin.
Romans 8:7 The mind governed by the flesh is hostile to God; it does not submit to God’s law, nor can it do so.
To those who say that it is impossible to fulfill God's Law, that is simply untrue. God was quite clear from the outset that it was within our grasp to keep the Law.
Deuteronomy 30:11-20 Now what I am commanding you today is not too difficult for you or beyond your reach. 12 It is not up in heaven, so that you have to ask, “Who will ascend into heaven to get it and proclaim it to us so we may obey it?” 13 Nor is it beyond the sea, so that you have to ask, “Who will cross the sea to get it and proclaim it to us so we may obey it?” 14 No, the word is very near you; it is in your mouth and in your heart so you may obey it.
15 See, I set before you today life and prosperity, death and destruction. 16 For I command you today to love the Lord your God, to walk in obedience to him, and to keep his commands, decrees and laws; then you will live and increase, and the Lord your God will bless you in the land you are entering to possess.
17 But if your heart turns away and you are not obedient, and if you are drawn away to bow down to other gods and worship them, 18 I declare to you this day that you will certainly be destroyed. You will not live long in the land you are crossing the Jordan to enter and possess.
19 This day I call the heavens and the earth as witnesses against you that I have set before you life and death, blessings and curses. Now choose life, so that you and your children may live 20 and that you may love the Lord your God, listen to his voice, and hold fast to him. For the Lord is your life, and he will give you many years in the land he swore to give to your fathers, Abraham, Isaac and Jacob.
Christ and his followers were quite clear that Love is the fulfillment of the Law. So central is love to all of this that God is himself said to be love.
Romans 13:8 Let no debt remain outstanding, except the continuing debt to love one another, for whoever loves others has fulfilled the law.
1 John 4:8 Whoever does not love does not know God, because God is love.
Matthew 7:21-23 Not everyone who says to me, ‘Lord, Lord,’ will enter the kingdom of heaven, but only the one who does the will of my Father who is in heaven. 22 Many will say to me on that day, ‘Lord, Lord, did we not prophesy in your name and in your name drive out demons and in your name perform many miracles?’ 23 Then I will tell them plainly, ‘I never knew you. Away from me, you evildoers!’
So then, through love we fulfill the Law and are known by God. On the other hand, those who do not love are not known by God - and these he casts out saying "I never knew you. Away from me, you evildoers."