Mr. Jaggers asks Pip to imagine that a woman who is being tried for murder has a child she has concealed from the world. She is obliged in her situation to tell her lawyer about the child, and the lawyer, coincidentally, knows a rich old lady who is searching for a child to raise as her own. The lawyer, pragmatic to the point of jadedness by his experiences, can see no better future for the child than that she be given up and put in the care of the old lady with means. The woman being tried follows the advice of her lawyer concerning the child, which turns out to be a fortuitous decision. The woman's mind is unhinged by her hardships and when she is acquitted she puts herself under the care of her lawyer; meanwhile, the child grows up and marries, never knowing that her real parents are alive.
The "imaginary" case that Jaggers "puts before" Pip is the true story of Magwitch, Molly, and Estella. Jaggers presents the case as hypothetical so that he will not have to admit to his part in the situation, even though he and Pip know very well about whom he is talking. Jaggers puts the case before Pip to emphasize his belief that no one would benefit from the secret being revealed now, and that things are better left alone (Chapter 51).
In Ch.51 Jaggers is for a moment taken aback when Pip confronts him with the truth that Magwitch is indeed Estella's father-something which he himself did not know. Then, Pip states "I wanted assurance of the truth from him."
Jaggers is a lawyer and his professional code of ethics prevents him from revealing the private and confidential information of his client Molly. Under normal circumstances he would have expressed his inability to reveal anything about Molly to Pip. But seeing Pip's pathetic condition-bandaged arm and his earnest determination; and with Pip appealing to Wemmick "Wemmick I know you to be a man with a gentle heart....I entreat you to say a word for me to Jaggers....to be more open with me," he undergoes a change of heart.
But he does not divulge the facts of Molly's background directly to Pip. He does so obliquely by presenting an "imaginary case" hypothetically ("put the case") even as he reveals the truth about Molly's background to Pip. At the end of Jaggers' account Pip says "I understand you perfectly" and agrees with Jaggers that Jaggers has not "admitted" (leaked out) the truth about Molly."
After murdering Magwitch's lover Molly sought Jaggers' help. At the same time Miss Havisham had asked Jaggers to get him a child to adopt. Jaggers agrees to help Molly if she hands over her child to him and Jaggers gives the child to Miss Havisham who brings her up as Estella.