This line is spoken by Mr. Rokesmith to Bella Wilfred in Our Mutual Friend. Bella was once a spoiled, insufferable snob who turned down Mr. Rokesmith's marriage proposal because she thought that he wasn't good enough for her. At this point in the novel, however, she has changed and become more humble and likable. She says to Mr. Rokesmith that she feels useless. When he tells her not to say that, she insists it is true. To comfort her, and because he is a good-hearted person who really believes what he is saying, Mr. Rokeford says to her,

No one is useless in this world who lightens the burden of it for anyone else.

This quote means that anyone who reaches out to help another person is doing useful work in the world, no matter what they might think of themselves or how trivial the help may seem. It is pure Dickens to see redemption in being good to our fellow man. The quote also characterizes Mr. Rokesmith as a man of compassion, worthy to marry Bella.

The quote by Charles Dickens, "No one is useless in this world who lightens the burden of it for anyone else," is timeless. It is just as relevant today as it was when Mr. Dickens wrote it in the 1800's. Basically, it means that as long as we are helping others in some way, we have a purpose in this world. When we lighten someone's burden, we make life easier for him/her in some way, so our lives are useful. This could be as simple as a smile for a person who is having a bad day, or it could be giving that person a kidney. Anything each of us can do to help somebody else  gives us purpose here in this world. In other words, our lives become meaningful when we help others.