This quote is very famous. The meaning of the quotation, however, would probably be different depending on the reader's perceptions.
I have always perceived this comment to mean that when someone is truly in love, and his or her actions are motivated only by love (which by definition is supposed to be unselfish and generous in nature), that a person in love never intentionally does things to hurt the object of that love. With this premise, Jenny Cavilleri Barrett is saying that the word "sorry" is never necessary because there is no ill intent on a true love's part, and apologies are never necessary.
Perhaps Jenny needs to say this because her husband Oliver was raised by a father who was a strongly imposing figure who expected perfection from others. And although Oliver may be like his father in some ways, it is important for Jenny to let him know that never making mistakes is not expected in the course of loving someone, and that there should be no regrets because true love is not about keeping score. It is about giving fully from one's heart out of love for the sake of love.