Jem does not fully understand the lesson that Atticus intends for him to learn from Mrs. Dubose. He does, however, realize that he is capable of dealing with difficult people, capable of keeping a level head in the face of abuse, and that all things come to an end.
Jem learns the meaning of Atticus's remark after Mrs. Dubose's death that he has wanted Jem to learn what real courage is,
It's when you know you're licked before you begin but you begin anyway and you see it through no matter what."
Atticus uses the example of Mrs. Dubose's fight against her morphine addiction as a teaching tool to show the children that it is not a man with a gun who is courageous, but a small, old lady who has the strength of character to fight even when she knows she is dying.
In addition, Atticus teaches the children about the virtue of charity as whenever Mrs. Dubose speaks insults about him, he ignores them; and, whenever he sees her as he traverses the street on his way home from his office, he tips his hat and speaks politely to her.
This is just another example of Atticus' advice about stepping into someone else's shoes and looking at things through their point of view before judging them. Atticus later stated that Mrs. Dubose was the bravest person he had ever known, and he wanted Jem to see that bravery doesn't just include heroic figures using their guns and fists. Jem also learned that punishments come in many varieties and often result in mixed feelings; only after Mrs. Dubose died did Jem learn the exact function of his presence and the depth of his emotions concerning her.
Jem learns the quote that Atticus makes in chapter three in reference to walking around in someone's skin before you judge. Atticus is trying to teach his children one of the most important lessons they can learn. "Judge not that you be not judged"(Matthew 7:1). It is one thing to think Ms. Dubose is a hateful woman, but it is another thing to realize she has health issues. Jem learns a valuable lesson through his contributions to another person who is hurting in more ways than one.
I think that it contributes to his becoming an adult because it teaches him not to judge people too quickly.
When Jem goes to see Mrs. Dubose, he learns that people who are really nasty in some ways can be admirable in others. This shows him that people are complex. They can have multiple sides to them within the same person. This kind of sophisticated thinking about other people is a pretty mature thing and learning it helps make him an adult.