" "It's really quite as indispensable as I say that Chad should be got back."
"Indispensable to whom? To you?"
"Yes," Strether presently said.
"Because if you get him you also get Mrs. Newsome?"
Strether faced it. "Yes."
"And if you don't get him you don't get her?"
It might be merciless, but he continued not to flinch. "I think it might have some effect on our personal understanding...."See in text(Book Third - I)
These lines of dialogue between Strether and his acute friend Waymarsh establish Strether's whole motivation for coming to Europe. He can marry his wealthy patroness Mrs. Newsome if he succeeds in persuading her son Chad to return to America. But if he does not succeed, he probably won't be able to marry her. He obviously has a small private income, but he can only feel secure at his late stage in life is he marries a wealthy woman.
The reader will realize later that Mme. de Vionnet wants to keep Chad in France because she is in love with him. So the entire novel is about a tug of war between Mrs. Newsome in America and Mme. de Vionnet in Europe, with Lambert Strether caught in the middle. It is really Mrs. Newsome, though never seen in person in the book, who is the protagonist, and Mme. de Vionnet who is the antagonist.