Firstly, what's the definition of a identity map? I've thought of one but don't know if it's correct :" $P$ is a identity map if $P(x)=x$ for all $x$ on its domain."
Second, how to prove a projection transformation is an identity map on a specific domain. More specifically,
Let $P: V \to V$ be a projection. Show that the restriction of $P$ to $\operatorname{range}(P)$ is the identity map on $\operatorname{range}(P)$.
Your definition of identity map is correct.
What did you try to solve the problem? Just start translating the data. Let $P:V\to V$ be a projection. What does that mean? It means that $\dots\dots$.
Now restrict $P$ to its range. So you have a map
\begin{align*} P_{\big|\text{range}\,P}:\text{range}\, P&\to\text{range}\,P\\ v&\mapsto P(v). \end{align*}
You want to show that $P_{\big|\text{range}\,P}$ is the identity map on $\text{range}\, P$, i.e, $$\forall v\in\text{range}\,P,P(v)=v.$$
Let's go. Let $v\in\text{range}\,P$. You want to compute $P(v)$. But wait a second: $v\in\text{range}\,P$. That's an important data. What does it mean? It means that $\dots\dots$.