The term cutting a covenant in Bible times had to do with something that Abraham experienced with God. Genesis 15. Which is the land covenant.
Explained here by Jan Markell's Olive Tree Ministries:
Cutting a Covenant – Olive Tree Blog
Animals were split in half (cut) and the terms were drawn up with the intent being that the one with greater power was demanding certain actions from the lesser one. If the lesser power failed in the contract they were to end up as dead as the animals cut in two were.
God took both parts by making Abraham sleep after cutting the animals. He cut the covenant with Abraham, but instead of Abraham passing thru the split animals- signifying Abraham taking responsibility to KEEP the covenant, God passed thru as the burning torch. This means that God lays on His own person the requirements of the covenant- HE keeps the covenant. It is an unconditional covenant.
And that unconditional covenant gave Abraham and his descendants the right to the land from the river of Egypt to the River Euphrates
17; And it came to pass, when the sun went down and it was dark, that behold, there appeared a smoking oven and a burning torch that passed between those pieces.
18 On the same day the Lord made a covenant with Abram, saying:
“To your descendants I have given this land,
from the river of Egypt to the great river, the River Euphrates—
Jesus own blood satisfied the demands of God's holiness- the atonement for our sins was paid by HIM on the cross. As
@Tall Timbers said, the cup is drunk in remembrance of that sacrifice.
What Copeland was getting at was similar to the Catholic doctrine of Trans substantiation- where they believe that the communion wine transforms into the blood of the Lord (and the wafer transforms into the body of the Lord).
Instead of that misunderstanding, if we look at the first century wedding we find another covenant- one of betrothal in which the hopeful would be groom offers a cup of wine to his prospective fiance and if she accepts his proposal, she drinks that cup of wine and they are now formally betrothed, which is far more than just engaged. They are as good as legally married, but he then has to go to his father's house to prepare a place for her which can take up to 2 years. If anyone asks the groom when the wedding is, he answers that he doesn't know, only his Father knows for sure. When the Father gives the command, the groom goes to get his bride and the wedding feast begins.
The price Jesus paid for His bride (us) was his own blood on the cross. When we take that cup we signify that we agreed to be His bride. We wait expectantly for His appearing to come and get us as our bridegroom.
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