Core Observation (Moving root changes distances)

When you move the root from node $u$ to its child $v$, every distance in the tree shifts. Nodes inside $v$'s subtree are now $1$ edge closer to the root. There are size[v] such nodes, so the total distance decreases by size[v].

Nodes outside $v$'s subtree are now $1$ edge farther from the root. There are n - size[v] such nodes, so the total distance increases by n - size[v].

Combining both: answer[v] = answer[u] - size[v] + (n - size[v]) = answer[u] + n - 2 * size[v]. This single formula lets you compute every node's answer from its parent's answer in $O(1)$.