Example 1 Let $f(x)$ be a periodic function defined on the set of real numbers with a period of 2, and it is an even function. When $x \in[2,3]$, $f(x)=-x$. Then when $x \in[-2,0]$, $f(x)=$ $\qquad$
When $x \in[-2,-1]$, $x+4 \in[2,3]$. By the periodicity of $f(x)$ with period 2, we have
$$
f(x)=f(x+4)=-x-4 \text {. }
$$

When $x \in[-1,0]$, $-x+2 \in[2,3]$.
By the fact that $f(x)$ is an even function with period 2, we have
$$
f(x)=f(-x)=f(-x+2)=x-2 \text {. }
$$

In summary, when $x \in[-2,0]$,
$$
f(x)=-3+|x+1| .
$$