Lemma. If a finite field $F$ has $p^{m}$ elements, then a polynomial $x^{p^{m}}-x$ in $F[x]$ factors in $F[x]$ as $x^{p^{m}}-x=\prod_{\lambda\in F}(x-\lambda)$
By the theorem I know that these polynomial has at most $p^{m}$ roots in $F$. But I don't know how to continue these proof. Could anyone help me with these, please?
This is the classification of finite fields: indeed, you want to prove that if a field $F$ has $p^m$ elements, then $$F\cong\{\textrm{zeros of }q_m(x)\},$$ where $q_m(x)=x^{p^m}-x\in F[x]$. This is true because:
$0^{p^m}=0$ and for every $\alpha\in F^\times$ (which is a finite cyclic group) one has $\alpha^{p^m-1}=1$, i.e. $\alpha^{p^m}-\alpha=0$. This proves that each of the $p^m$ elements of $F$ is a root of $q_m(x)$.
The degree of $q_m(x)$ is $p^m$. Hence all of its roots are those found in point 1: they are the elements of $F$. This means that $q_m(x)$ splits as you wrote in your question.