Vocabulary - Subtree

A subtree rooted at node $v$ includes $v$ and all nodes below it. When you compute $dp[v]$, you only need information from $v$'s subtree. This is why tree DP works: each subtree is independent.

Once you know the answer for every child's subtree, you can compute the answer for the parent. Leaves have trivial subtrees (just themselves), so they're your base cases. If you try to compute $dp[v]$ before computing $dp[\text{child}]$, you'll be missing the values you need. The order matters.