Moses received the Law from the Lord Almighty at Sinai, and he was the mediator/law giver to the Children of Israel.
Exo 19:3 And Moses went up unto God, and the LORD called unto him out
of the mountain, saying, Thus shalt thou say to the house of Jacob,
and tell the children of Israel; Exo 19:4 Ye have seen what I did
unto the Egyptians, and how I bare you on eagles' wings, and brought
you unto myself. Exo 19:5 Now therefore, if ye will obey my voice
indeed, and keep my covenant, then ye shall be a peculiar treasure
unto me from among all peoples: for all the earth is mine: Exo 19:6
and ye shall be unto me a kingdom of priests, and an holy nation.
These are the words which thou shalt speak unto the children of
Jesus the Messiah, of Whom Moses was a type, ascended upon a mountain and provided the Law of Christ to His disciples.
Mat 5:1 And seeing the multitudes, he went up into the mountain: and when he had sat down, his disciples came unto him:
Gal 6:1 Brethren, even if a man be overtaken in any trespass, ye
which are spiritual, restore such a one in a spirit of meekness;
looking to thyself, lest thou also be tempted. Gal 6:2 Bear ye one
another's burdens, and so fulfill the law of Christ.*
There are a few contrasts between the covenant at Sinai and the New Covenant of Christ we should notice. First, the covenant at Sinai found its fulfillment in Christ. Christ, through His blood, enacted a better covenant--one with better promises.
Heb 8:6 But now hath he obtained a ministry the more excellent, by
how much also he is the mediator of a better covenant, which hath been
enacted upon better promises.
As implied in the previous verse, the New Covenant is different in purpose and in scope from the old.
What we find in Matthew 5 is not a new set of written laws, but an expounding of a law written on the believer's heart. It has nothing to do with the rigidness of the Mosaic Law (obey and live, disobey and die), It has everything to do with yielding to Christ because of a changed heart and our desire to submit to His kingdom rule.
Heb 10:16 This is the covenant that I will make with them After
those> days, saith the Lord; I will put my laws on their heart,
And upon their mind also will I write them;
Second, the mountain of the Lord was not approachable to those who were to adhere to the covenant law in Moses' day (see Ex 19:12). But Jesus, our New Law Giver, spoke to His own directly. As the writer of Hebrews wrote, "(God) has spoken to us in his Son" (Heb 1:2). In this we see how God's children are welcomed into the throne of grace by the blood of His beloved Son (Heb 4:16).
Heb 12:18 For ye are not come unto a mount that might be touched, and
that burned with fire, and unto blackness, and darkness, and tempest,
Heb 12:19 and the sound of a trumpet, and the voice of words; which
voice they that heard entreated that no word more should be spoken
unto them: Heb 12:20 for they could not endure that which was
enjoined, If even a beast touch the mountain, it shall be stoned; Heb
12:21 and so fearful was the appearance, that Moses said, I
exceedingly fear and quake: Heb 12:22 but ye are come unto mount
Zion, and unto the city of the living God, the heavenly Jerusalem, and
to innumerable hosts of angels, Heb 12:23 to the general assembly
and church of the firstborn who are enrolled in heaven, and to God the
Judge of all, and to the spirits of just men made perfect, Heb 12:24
and to Jesus the mediator of a new covenant, and to the blood of
sprinkling that speaketh better than that of Abel.
I hope all this answers your question. If so, I advise a book titled: In Defense of Jesus, the New Lawgiver, by John Reisinger.