Moses was called to deliver the Israelites from the hand of Pharaoh in the Egyptians civilization. WHO WERE THE GODS OF ANCIENT EGYPT THEN?
Amun - The biggest God of Egypt who was God of the royal family. Ra - God of the sun. Hapi - God of the Nile. When you meet these gods that are connected to external sources, they probably present themselves like "I am Amun, I am the god of the royal family", "I am Ra, I am the god of the sun", "I am Hapi, I am god of the Nile" showing that they are inferior to God almighty because they are connected to external sources. Now, imagine you are in Moses shoe where you have to present God to the Israelites so they will know that they can be free. Imagine you are an Israelite and you live in that Egyptian civilization and you might have probably heard those gods, about the Sun god and how people are sacrificing to him, about the Nile, about how powerful he is, about the god that protects the royal family, etc. So when Moses asked God "what should I tell them your name is?" just like Amun who would say "I am Amun, god of the royal family", Ra will say "I am Ra, God of the sun", Hapi will say "I am god of the Nile", God said "Tell him that I AM THAT I AM" meaning my power is in myself, I am self sufficient, I control everything that these gods are powered by. I am the God of it all.
HOW DID GOD PROVE THIS?
He proved it by the plagues. This is what he did. He told Moses to tell Pharaoh "Let my people go" and Pharaoh was like "Who does he think he is?" , God struck with the plagues. So in order for God to prove that he is more powerful than God Nile, he told Moses to turn the Nile into blood. Pharaoh sees this. "So the God is more powerful than Nile". And then, we have Ra the god of the sun (People still worship Ra till today) what did God do? He darkened the sun for three days. Pharaoh sees this "So he is more powerful than Ra". And then the last one Amun the protector of the royal family. God killed all the first born sons specifically Pharaoh's. If God could have killed all the first borns and Pharaoh's son is still alive, people will have said "Amun is still powerful than God cause he was able to protect Pharaoh's son but he killing all first born sons including pharaoh's shows he is more powerful than Amun. God just defeated the Big Three! That's why Pharaoh let the Israelites go because he was probably in fear. This God just beat every god that is connected to everything in their society.
The name "I Am THAT I AM" might be connected to thousands of things due to the book of revelations but it shows how vast God is. God works wisely and what he does can actually have thousands of meanings of which when put together will make sense!
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A MUST READ: HOW GOD PROVED HIMSELF AGAINST THE GODS OF EGYPT