What does “I am that I am” mean when God says it to Moses?

Answer 1

The name Yahweh, I am that/who I am, is quite fluid in meaning and application:

1) As a name for God, it is quite brilliantly an attack against the idolatry of the nations. For God to declare that he is necessarily then denotes that all other are not. Considering the thematic backdrop of the Exodus is a battle between Yahweh and the gods of the Egyptians, it makes sense why God would choose a name which both reveals who he is while simultaneously rejecting the existence of other deities.

2) The meaning of "Yahweh" has been debated over the years, but most scholars agree that the verb is non-tensive in form and relates to existence. Thus, it is possible that by God, in declaring that he is/exists, is declaring his internality (cf. Rev. 1 adds the phrase "who was, and is, and is to come" perhaps as a means of fully supplying this meaning).

3) Yahweh reveals a covenantal connection with the people of Israel. This is a theme drawn out in other places within the OT, but is especially evident in the Prophets. The fact that this name is first given in connection with Abraham, Isaac, and Jacob, reveals that the name is given not as a general name, but a name directly tied to the Patriarchs and their descendants. Hosea 1:9 draws on this idea when God commands Hosea to name Gomer's child "Not my people," only to then directly state, " I am [Yahweh] not your God." The fact that Hosea employs Yahweh here is important because it designated a covenantal break between the people of Israel and God.

~ Cameron Brock

Answer 2

He is only implying that he's known him. He's the only God that Moses and his family before him has known since he created Adam. Up to this point God finally states his name to the Israelites. Their name that they can call him by.