What does Genesis 21:1 mean?

The Lord keeps His promises, no matter how unlikely they may seem from a human perspective. Not only had God promised to make Abraham a great nation, He had promised specifically that this nation would come through Sarah, a vow made twice within the last year (Genesis 17:16–21; 18:10–15). Now the Lord visits Sarah "as he had said" and the Lord does to her "as he had promised." Somehow, in this visiting, the Lord made Sarah, finally, able to conceive a child with Abraham.

It's interesting that, though most of God's promises about the great nation that would flow from Abraham have been made directly to Abraham, the focus of the promise's fulfillment rests on Sarah. God maintains a relationship with her and His promise is made to and through her.

The repetition and phrasing in this verse is meant to slow us down and cause us to notice that God is faithful. He does as He says.