God says to Moses "I am who I am," in response to Moses. Why does God respond to Moses in a cryptic fashion fashion, and what is the specific meaning of God's words?
This is a great question. The full account of this passage is in Exodus 3. It is a good idea to read the whole chapter, because it will give you the context of the name of God. With this said, let me make a few points.
First, what is cryptic to us is not necessarily cryptic to Moses or we can say even the original readers of the text. As you read the narrative, there is no sense of confusion for Moses. He simply takes it as the name of God.
Moses said to God, "Suppose I go to the Israelites and say to them, 'The God of your fathers has sent me to you,' and they ask me, ‘What is his name?’ Then what shall I tell them?" Exodus 3:13
Second, when God reveals himself as "I am," there is logic to it. God is saying that he simply is. In other words, he is the great being who always existed and always will. In short, he is. He is the one who is the same yesterday, today and tomorrow. This philosophical and theological explanation goes a long way in trying to understand who God is.
I will add a link about the whole exodus narrative, which will give you the main context of the book of Exodus.