Let $(M,g)$ be spacetime and consider the space of solutions to the Klein-Gordon equation endowed with the inner product:

$$(\phi_1,\phi_2)=\int_\Sigma (\phi_1\nabla_\mu \phi_2^\ast-\phi_2^\ast\nabla_\mu \phi_1) n^\mu \sqrt{|\gamma|}d^{n-1}x$$

Let $\{\phi_i\}$ be a complete set of solutions on this space and $K$ a timelike Killing vector field on $(M,g)$.

If I understood well (and I might have not), in his book Spacetime and Geometry, Sean Carroll says that $\phi_i$ is a positive frequency solution to the KG equation if there is $\omega \in(0,\infty)$ such that

$$\mathscr{L}_K \phi_i=-i\omega \phi_i,$$

and it is a negative-frequency solution if

$$\mathscr{L}_K \phi_i=i\omega \phi_i.$$

Now why does one need a Killing field to judge this? Why this can't be done with an arbitrary timelike vector field?

The Lie derivative of a $C^\infty(M)$ function is just the directional derivative of the function along the integral lines of the field.

Why this condition says that a function is a positive or negative frequency, and why we need the Killing field?