In Chapter 44 of "Great Expectations," Pip calls upon Estella after learning much of her past from Provis (Magwitch) who has returned from New Zealand in search of Pip, wishing to tell Pip that it is he who is his benefactor.  After Pip goes to Mrs. Brandley's in Richmond, Estella's maid informs him that Estella has gone to the country.

Next day I had the meanness to feign that I was under a binding promise to go down to Joe; but I was capable of any meanness to Joe or towards his name.

When Pip stops at the Blue Boar instead of going to Joe's, he encounters the brutish Drummle, whom he learns is on his way to visit Estella, as well.  The next day Pip himself visits Saris House and finds Estella seated on a cushion near Miss Havisham.  Pip tells Miss Havisham,

It will not surprise you; it will not displease you.  I am as unhappy as you can ever have meant me to be....I have found out who my patron is.  It is not a fortunate discovery, and is not likely ever to enrich me in reputation station, fortune, anything....I suppose I did really come her...as a kind of servant, to gratify a want or a whim, and to be paid for it?

Miss Havisham, then, affirms his question, taking no blame for her actions.  Pip asks her one favor for her relative, Herbert:  He requests some money be give secretively since Pip is to blame for the failing of his and Herbert's business venture, having been the one to initiate it.  Pip, then, turns to Estella, saying that he had hoped that Miss Havisham meant for him and Estella to marry.  Though Estella is unmoved, Miss Havisham puts her hand to her heart.  In answer to Pip's inquiries about Bentley Drummle, Estella informs him that she is to be married to the man.  Pip declares his eternal love for Estella, kisses her hand, and leaves noticing that Miss Havisham still has her hand covering her heart, in "a ghastly stare of pity and remorse."

The night porter hands Pip a mysterious note from Wemmick:  "Don't go home."  This note evidences the great friendship of Wemmick in contrast to the self-serving motives of Miss Havisham and Estella in their past actions towards Pip.

Why does Pip want to go back home?  What does he find when he visits Satis House?  

I assume you are talking about the end of the book.  Pip wants to go home and talk to Joe.  He wants to ask him for forgiveness for how he's treated him.  He also wants to repay Joe for all of his bills.  He's finally matured and realizes how he's treated Joe since he's come into his "great expectations."  While he's home, he wants to propose to Biddy.  She's the one who was always good, always pure, and he should have seen it long ago.  Those are the reasons he wants to return home.

Then in the last chapter he returns to Satis House and finds Estella there.  He finds that she has been all but broken by Drummle.  But he had an accident and died.  She and Pip decide to never part again (in the second ending--the one I like better).