How does Jaggers explain his role in giving Estella to Miss Havisham?

He simply is the agent for giving the child to Miss Havisham; he knows that Miss Havisham has no children and that she has been "left at the altar" and has nothing else to concentrate on.

From the text:

"Put the case, Pip, that here was one pretty little child out of the heap, who could be saved; whom the father believed dead, and dared make no stir about; as to whom, over the mother, the legal adviser had this power: "I know what you did, and how you did it. You came so and so, this was your manner of attack and this the manner of resistance, you went so and so, you did such and such things to divert suspicion. I have tracked you through it all, and I tell it you all. Part with the child, unless it should be necessary to produce it to clear you, and then it shall be produced. Give the child into my hands, and I will do my best to bring you off. If you are saved, your child is saved too; if you are lost, your child is still saved." Put the case that this was done, and that the woman was cleared."