If you're wondering where this language came from, you can look at Genesis 15. Here God makes a covenant promise to Abraham, that his descendants would be like the stars in the sky or the sand on the sea, and that He will give him the land that He promised to him. But how could Abraham know that God would keep His promise, especially since Abraham was an old man with no hope of having a child?
In Abraham's time there was a common ceremony that a king would use to enter into a covenant relationship with people he had conquered. The conquered people had to take animals from their flocks or herds and cut them in two, laying the halves on the ground with a path between them. The king would then make a promise about what he would do for them, and he would also command the people what they had to do for him, like the tribute they had to pay. He would then command the people to walk between the animals that had been cut in two, and the king would threaten to do the same thing to the conquered people if they would not obey his commands and keep the covenant. This is how the language of "cutting a covenant" came about.
When God made His promises to Abraham, He told him to do the same thing. God said: "Bring me a heifer, a goat and a ram, each three years old, along with a dove and a young pigeon." Abraham did as God commanded, cut the animals in two and laid them out on the ground. Then he waited. I'm sure that Abraham was expecting God to tell him what he had to do as a beneficiary of the covenant promises that God had made to him. And I'm sure He expected God to command him to walk the path between the pieces as a threat of what would happen to him if he did not do as God commanded.
As Abraham sat there driving off the birds of prey as he waited for God's command, something entirely different happened. "When the sun had gone down and it was dark, behold, a smoking fire pot and a flaming torch passed between these pieces." What was that all about? The next verse says, "On that day the LORD made a covenant with Abraham." Essentially, in the form of "a smoking fire pot and a flaming torch" God Himself passed between the cut pieces of animal that were laying on the ground. Instead of threatening Abraham, God threatened Himself. It was as though God told Abraham, "If I fail to keep the promises that I have made to you this day, I will destroy myself." How amazing is that?
God essentially told Abraham that nothing would keep His promises from coming true. Sadly, however, Abraham's descendants broke the covenant that God made with them. They forsook God and worshiped and served the idols of their pagan neighbors. What would God do now? Would He abandon them or destroy them? In fact, what God did was make a new covenant, a covenant that promised "I will forgive their wickedness and remember their sins no more." (Jeremiah 31:34) But how could God do that? Only through the death of His own Son, Jesus, could God offer the forgiveness of sins freely to all who believe as Abraham did.

Who can comprehend such a God, a God would would let nothing, nothing interfere with His promises, not even if it meant the death of His own Son. What an amazing God we serve! He cut a covenant with Abraham, and kept it by cutting down His own Son for our sins. Praise God for His all surpassing grace!