The first item that is found is chewing gum; Scout is alerted to its presence by some tinfoil, "sticking in a knot-hole just above my eye level, winking at me in the afternoon sun." She takes the gum and chews it, and Jem notices that she has it. Gum is quite the thing to kids, and Boo knew enough to know that she would enjoy it, and to catch her attention with the tin foil in the first place. So, it shows that he has an understanding of children, and of Scout's daily route past his house. Boo also cares a lot for the children; he gives them very valuable indian-head pennies, all shined up and wrapped like a present. Later, they find two soap sculptures, that "were almost perfect miniatures of two children" that happen to look just like Boo and Scout. This shows that he cares a lot for them, and spends quite a bit of time watching their play games, because he knows, in detail, what they look like. The last thing that they find is a pocketwatch, and that indicates that Boo wants to give them valuable things, and to show that he cares.
Overall, the gifts reveal Boo to be a friendly, sensitive man who knows what kids like very well, cares deeply for Boo and Scout, and knows them and their activities. It's an interesting relationship that he tries to forge with the kids, and his feeling of connection to them is going to come into play in a much larger way later in the novel.
One item in the tree that has great significance and allows insight into the character of Arthur Radley is the broken watch. The watch symbolizes that Arthur Radley--the respectful, kind, sensitive young man--is stunted in his maturation process, or his coming of age tale, by an overbearing religious zealot of a father. Each item in the tree demonstrates the potential that was ultimately wasted. Arthur reaches out to children because they are the only ones who can truly understand him because they are about the same mental age. The tragedy of Arthur Radley is that even after the father's death his brother becomes his keeper, which reinforces the religious overtones, and continues the isolation established by the father and enabled by the town. After years of forced seclusion Arthur finally finds and seeks to communicate with others, but his attempts are thwarted by his brother, his captor, thereby forcing him to retreat back into the shadows for a time and keeping him essentially as a child.