Dill Harris has been through a lot for a kid "going on seven."  He's been sent away from home to live with his Aunt Rachel, and doesn't have a father.  When Scout brings this up, he simply lowers his head, but Jem quickly corrects her rudeness.  Unaware that some children have fathers who have left the family, Scout cannot comprehend the level of discomfort in the situation, although Jem has a better idea.

Later on, Dill sneaks on a train to come visit Scout and Jem, traveling to Maycomb, Alabama from Meridian, Mississippi, all by himself.  This shows that he has some maturity.  Not to mention the fact that he's seen so many movies at the theatre using the money he won from the "beautiful child contest" in Meridian.

The "I'm little but I'm old" line says a lot about his life experiences, even though he's under the age of seven at the time we are introduced to him early in the novel.