Jesus continues to speak ex ousia (with authority). Leading with "Amen," He declares the profound truth of what will follow. At the very beginning of the passage today, Jesus says that whoever keeps His word will never see death. Who else but God could speak with authority over life and death? Unlike the prophets, Jesus does not bring His message with "Thus saith the Lord." Rather, it is more like "Thus saith me, for I am the Lord." None but God has ultimate power over life and death, and none but God could speak as Jesus does here. Jesus also speaks of the doxa (glory) that He has from the Father. The divine splendor is integral to the Person and work of Christ. He has it intrinsically, from within His very being.
All of this helps contextualize the "mic drop" moment, when He later says "...before Abraham was, I AM." There was no mistaking this claim to divinity. This so incensed the people that they wanted to kill Him. In this, they would later succeed, albeit only for a time.
What so disrupts the thought pattern of Jesus' audience is that no man could be God. That much is beyond dispute. It is true that no mere man, that was only man, could be God. However, human conceptions and limitations do not apply to God. Those who read their Hebrew Scriptures should have been keener to the fact that Yahweh was constantly turning the expectations of His people upside down. He was constantly causing them to expand their view of Him, to not 'box in', as it were, their conception of the divine. Think about the way God brought Israel out of Egypt, led them into the Promised Land, gave them victory over their enemies, set up a place of worship and divine dwelling in the Temple, and so forth. None of these would have been speculated by mere human imaginings. The people had grown so used to making God in their own image they forgot that it was the other way around.
Jesus directly confronts this presumption and the attending spiritual malaise. It is not what we would think of as a cordial exchange. When light confronts darkness, the darkness will try to fight against the light. But the light will not be overcome (John 1:1-18).