Another viewpoint - a Killing transformation is a coordinate transformation that preserves the functional form of the metric, i.e. ##g'(x') = g(x)##. Under the coordinate change ##x' = x + \xi## (or ##x = x' - \xi##),\begin{align*}
g'_{ab}(x') &= \frac{\partial x^c}{\partial x'^{a}} \frac{\partial x^d}{\partial x'^{b}} g_{cd}(x) \\
&= (\delta^c_a - \partial_a \xi^c)(\delta^d_b - \partial_b \xi^d) g_{cd}(x) \\
&= g_{ab}(x) - g_{ad}(x) \partial_b \xi^d - g_{cb}(x) \partial_a \xi^c + O(\xi^2)
\end{align*}To first order in ##\xi## the Taylor expansion of the LHS is \begin{align*}
g'_{ab}(x') = g'_{ab}(x+\xi) = g'_{ab}(x) + \xi^e \partial_e g'_{ab}(x)
\end{align*}so to first order in ##\xi##,\begin{align*}
g'_{ab}(x) + \xi^e \partial_e g'_{ab}(x) &= g_{ab}(x) - g_{ad}(x) \partial_b \xi^d - g_{cb}(x) \partial_a \xi^c \\
\implies \xi^e \partial_e g_{ab}(x) &= - g_{ad}(x) \partial_b \xi^d - g_{cb}(x) \partial_a \xi^c
\end{align*}since ##g'_{ab}(x) = g_{ab}(x)## from the Killing condition. Let's do some work on the RHS,\begin{align*}
- g_{ad} \partial_b \xi^d - g_{cb} \partial_a \xi^c &= -\partial_b \xi_a - \partial_a \xi_b \\
&= -D_b \xi_a - \Gamma^c_{ab} \xi_c - D_a \xi_b - \Gamma^c_{ba} \xi_c \\
&= -(D_b \xi_a + D_a \xi_b) - 2\Gamma^c_{ab} \xi_c
\end{align*}Remember that\begin{align*}
\Gamma^c_{ab} \xi_c &= \frac{1}{2} \xi_c g^{cd}(\partial_a g_{db} + \partial_b g_{ad} - \partial_d g_{ab}) \\
&= \frac{1}{2} \xi_c(\partial_a \delta^c_b + \partial_b \delta^c_a - \partial^c g_{ab}) \\
&= \frac{1}{2} \xi_c (0 + 0 - \partial^c g_{ab}) \\
&= -\frac{1}{2} \xi^c \partial_c g_{ab}
\end{align*}so that ##-2\Gamma^c_{ab} \xi_c = \xi^c \partial_c g_{ab}##, and overall\begin{align*}
\xi^e \partial_e g_{ab} &= -(D_b \xi_a + D_a \xi_b) + \xi^c \partial_c g_{ab}
\end{align*}which gives you ##D_b \xi_a + D_a \xi_b = 0##.