What does this line this mean in Pride and Prejudice by Jane Austen? "I do not know how you will ever make him amends for his kindness."
The line to which you refer is found in the second chapter of Pride and Prejudice by Jane Austen. In this chapter, Mrs. Bennet and her daughters are discussing their new, rich neighbor, Mr. Bingley. They are all eager to meet the new, rich, eligible bachelor at the soonest opportunity; however, propriety states that the girls cannot just introduce themselves. Time is of the essence for Mrs. Bennet and at least some of the girls, for they do not want any of the other girls in the neighborhood to make the rich Mr. Bingley's acquaintance before them.
The women are in despair of Mr. Bennet ever doing his husbandly and fatherly duty by going to make the proper introductions so they can begin trying to trap the rich young man into marrying one of the Bennet girls. But Mr. Bennet surprises them and says he visited Bingley (did I mention he is rich?) just that morning; then he leaves the room to avoid the reaction he knows will inevitably come.
That is when Mrs. Bennet says the following:
"What an excellent father you have, girls,'' said she, when the door was shut. "I do not know how you will ever make him amends for his kindness; or me either, for that matter. At our time of life, it is not so pleasant I can tell you, to be making new acquaintance every day; but for your sakes, we would do any thing."
The portion of the quote you asked about can be roughly translated this way: "I have no idea how you will ever be able to pay your father back for this great act of kindness he did for you."