Why is $\chi_{[\mu}\nabla_\nu \chi_{\sigma ]} = 0$ at the Killing horizon? Let $\chi$ be a Killing vector field that is null along a Killing horizon $\Sigma$
Why is $\chi_{[\mu}\nabla_\nu \chi_{\sigma ]} = 0$ at $\Sigma$?
 A: This is a partial answer. It assumes that the Killing field $\chi$ is normal to the Killing horizon. This implies that $\chi$ is null along the horizon, so it is consistent with the condition given in the OP. However, a Killing field that is null along the horizon is not necessarily orthogonal to the horizon; that's why this is only a partial answer.
Given a function $\Phi$ on spacetime and a constant $c$, the equation $\Phi=c$ defines a hypersurface. The vector field 
$$
n_a=\partial_a\Phi=\nabla_a\Phi
\tag{1}
$$
is orthogonal to that hypersurface. Now, if $\chi_a$ is any vector field orthogonal to this family of hypersurfaces, we must have 
$$
\chi_a=mn_a
\tag{2}
$$
for some function $m$. If $\chi$ is normal to the Killing horizon, then it can be written in the form (2) along the horizon.
Equation (2) implies
$$
   \nabla_b\chi_a=\nabla_b(m\nabla_a\Phi)
=
(\nabla_b m)(\nabla_a\Phi)
+
m\nabla_b\nabla_a\Phi.
\tag{3}
$$
Now consider $\chi_{[c}\nabla_b \chi_{a ]}$, where the square brackets denote complete antisymmetrization (which is what I'm assuming they mean in the OP). According to the preceding equations, 
$$
\chi_{[c}\nabla_b \chi_{a ]}
=
m(\nabla_{[c}\Phi)(\nabla_b m)(\nabla_{a]}\Phi)
+
m^2\nabla_{[c}\Phi\nabla_b\nabla_{a]}\Phi.
\tag{4}
$$
The first term in (4) is zero because of the identity
$$
(\nabla_{[c}\Phi)(\nabla_{a]}\Phi) = 0,
\tag{5a}
$$
and the second term in (4) is zero because of the identity 
$$
\nabla_{[b}\nabla_{a]}\Phi = 0
\hskip2cm
\text{(zero torsion)}.
\tag{5b}
$$
 Altogether, this shows that
$$
\chi_{[c}\nabla_b \chi_{a ]} = 0
\tag{6}
$$
whenever the vector field $\chi$ is orthogonal to the given hypersurface. This is (part of) the Frobenius theorem.
This derivation of equation (6) is outlined in section 2.3.3 in Poisson (2002), "An advanced course in general relativity," https://www.physics.uoguelph.ca/poisson/research/agr.pdf.
