Example 3 Find the range of $y=\sqrt{x^{2}-2 x+5}+\sqrt{x^{2}-6 x+13}$.

Analysis: If points $A$ and $B$ are placed above the $x$-axis, then $AB$ is parallel to the $x$-axis, so we can solve the problem by placing the points below the $x$-axis.
Solution: The original expression is $y=$ $\sqrt{(x-1)^{2}+(0-2)^{2}}+$ $\sqrt{(x-3)^{2}+(0+2)^{2}}$, with moving point $P(x, 0)$, and fixed points $A(1,2), B(3,-2)$. Connecting $A B$ intersects the $x$-axis at $P(2,0)$, as shown in Figure 7.

When $x=2$, $y_{\text {min }}=|A B|=2 \sqrt{5}$,
$\therefore$ the range is $[2 \sqrt{5},+\infty)$.