Solution:
Put $x - y = r$, $y - z = s$. Then $z - x = - (r + s)$, and
$$\left(x - y\right)^{5} + \left(y - z\right)^{5} + \left(z - x\right)^{5} = r^{5} + s^{5} - (r + s)^{5}$$
Expand $(r + s)^5$:
$$(r + s)^5 = r^5 + 5r^4s + 10r^3s^2 + 10r^2s^3 + 5rs^4 + s^5$$
So,
$$r^5 + s^5 - (r + s)^5 = r^5 + s^5 - [r^5 + 5r^4s + 10r^3s^2 + 10r^2s^3 + 5rs^4 + s^5]$$
$$= -5r^4s - 10r^3s^2 - 10r^2s^3 - 5rs^4$$
$$= -5rs(r^3 + 2r^2s + 2rs^2 + s^3)$$
$$= -5rs(r^2 + rs + s^2)(r + s)$$
Therefore,
$$\left(x - y\right)^{5} + \left(y - z\right)^{5} + \left(z - x\right)^{5} = -5rs(r + s)(r^2 + rs + s^2)$$
Recall $r = x - y$, $s = y - z$, $r + s = x - y + y - z = x - z = -(z - x)$.

Thus, $rs(r + s) = (x - y)(y - z)(z - x)$ (up to sign), and the expression is divisible by $5(x - y)(y - z)(z - x)$.