I will only give a sketch of the argument. We will use induction on the number $n$ of vertices of $T$. When $n = 1$ the claim is clear. Suppose then that it hold for trees of size $n$. Let $T$ be a tree of size $n+1$. Pick a leaf $v$ of $T$. (This is where we really use the fact that $T$ is a tree.) Let $C_1,C_2$ be proper $3$-colorings of $T$. Their restrictions to $T - \{v\}$ are also proper $3$-colorings and hence by induction we get that there exists a path between them. Because $v$ was a leaf, this path also induces a path between $C_1$ and $C_2$. More precisely we will first get a path from $C_1$ to some proper $3$-coloring of $T$ which agrees with $C_2$ on $T-\{v\}$. (This uses the fact that $v$ is a leaf, because we can modify the color of $v$ when it is necessary.) Because $v$ was a leaf, sucha coloring is either $C_2$ or adjacent to it.

I will only give a sketch of the argument. We will use induction on the number $n$ of vertices of $T$. When $n = 1$ the claim is clear. Suppose then that it holds for trees of size $n$. Let $T$ be a tree of size $n+1$. Pick a leaf $v$ of $T$. (This is where we really use the fact that $T$ is a tree.) Let $C_1,C_2$ be proper $3$-colorings of $T$. Their restrictions to $T - \{v\}$ are also proper $3$-colorings and hence by induction we get that there exists a path between them. Because $v$ was a leaf, this path also induces a path between $C_1$ and $C_2$. (This uses the fact that $v$ is a leaf, because we can modify the color of $v$ when it is necessary.)