We use here the shortened notation: transfer the prime that denotes components of a tensorial quantity in the primed frame of reference to the indices ${A'}_{\nu}\equiv A_{\nu'}$, ${x'}^{m}\equiv x^{m'}$, and denote derivatives with a comma $\partial_{\mu}{T^{\ldots}}_{\ldots}={T^{\ldots}}_{\ldots\,,\mu}$ and semicolon $\nabla_{\mu}{T^{\ldots}}_{\ldots}={T^{\ldots}}_{\ldots\,;\mu}$.
The transformation rule for ${\Gamma^{\mu}}_{\nu\lambda}$ is fixed by the condition that for arbitrary vector $A_\mu$ its covariant derivative $A_{\mu;\nu}$ is a tensor:
\begin{align*}
A_{i';k'}=&
{x^m}_{,i'}{x^n}_{,k'}A_{m;n}=
{x^m}_{,i'}{x^n}_{,k'}
\left({A_{m}}_{,n}-
{\Gamma^{p}}_{mn}A_{p}\right)=\\
&={x^m}_{,i'}{x^n}_{,k'}{A_{m}}_{,n}-
{x^m}_{,i'}{x^n}_{,k'}\;
{\Gamma^{p}}_{mn}A_{p}\;;\\
A_{i';k'}=&A_{i',k'}-{\Gamma^{p'}}_{i'k'}A_{p'}=
\left({x^{j}}_{,i'}A_{j}\right)_{,k'}-
{\Gamma^{p'}}_{i'k'}\;{x^{q}}_{,p'}A_{q}=\\
&={x^{j}}_{,i',k'}A_{j}+{x^{j}}_{,i'}{x^{n}}_{,k'}A_{j\,,n}-
{\Gamma^{p'}}_{i'k'}\;{x^{q}}_{,p'}A_{q}=\\
&={x^{p}}_{,i',k'}A_{p}+
{x^{m}}_{,i'}{{x^{n}}_{,k'}}A_{m\,,n}-
{x^{p}}_{,q'}{\Gamma^{q'}}_{i'k'}\;A_{p}.
\end{align*}
Equating the right hand sides and taking into account that $A^\mu$ is arbitrary, we have
\[{x^p}_{,q'}{\Gamma^{q'}}_{i'k'}=
{x^{m}}_{,i'}{x^{n}}_{,k'}\;{\Gamma^{p}}_{mn}A_{p}
+{x^{p}}_{,i',k'}.\]
Contracting with ${x^{j'}}_{,p}$ and taking into account that ${x^{j'}}_{,p}{x^{p}}_{,q'}=\delta^{j'}_{q'}$, we finally get
\begin{equation}\label{GammaTranform}
{\Gamma^{j'}}_{i'k'}={x^{p}}_{,i',k'}\,{x^{j'}}_{,p}+
{x^{j'}}_{,p}{x^{m}}_{,i'}{x^{n}}_{,k'}
\;{\Gamma^{p}}_{mn},
\end{equation}
or in expanded notation
\begin{equation}\label{GammaTransformFull}
{{\Gamma'}^{\,j}}_{ik}=
\frac{\partial^{2}x^{p}}
{\partial {x}^{\,i'}\partial {x}^{\,k'}}
\frac{\partial x'^j}{\partial x^p}
+\frac{\partial {x}^{\,j'}}{\partial x^{p}}
\frac{\partial x^{m}}{\partial {x}^{\,i'}}
\frac{\partial {x}^{\,n}}{\partial {x}^{\,k'}}
{\Gamma^{p}}_{mn}.
\end{equation}
Let the value of $\Gamma^{i}_{kl}$ at a given point $X$ be $\tilde{\Gamma}^{i}_{kl}=\tilde{\Gamma}^{i}_{lk}$. Then we can move the coordinate origin to $X$ and make the following coordinate transformation:
\begin{align}\label{LocalGeodesic}
x^{i'}&=x^{i}-{\textstyle\frac{1}{2}}
\tilde{\Gamma}^{i}_{kl}x^{k}x^{l};\\
{x^{i'}}_{,n}&=\delta^{i}_{n}-
\tilde{\Gamma}^{i}_{kn}x^{k};\nonumber\\
{x^{i'}}_{,n,m}&=-\tilde{\Gamma}^{i}_{mn}.\nonumber
\end{align}
At the point $x=x'=0$ then ${x^{i'}}_{,j}=\delta^{i}_{j}$, and therefore ${x^{i}}_{,j'}=\delta^{i}_{j}$, so the considered transformation preserves tensor components at the origin.
Substituting the derivatives into the transformation law for $\Gamma$ (\ref{GammaTransformFull}), we obtain the connection coefficients in $X$ in the new reference frame:
\[\tilde{\Gamma}^{i'}_{k'l'}=
\delta^{m}_{k'}\delta^{n}_{l'}\delta^{i'}_{p}
\tilde{\Gamma}^{p}_{mn}-
\delta^{i'}_{p}\tilde{\Gamma}^{p}_{k'l'}=
\tilde{\Gamma}^{i'}_{k'l'}
-\tilde{\Gamma}^{i'}_{k'l'}= 0.\]
Thus for any given point $X$ we can always bring the metric in the point to the diagonal form and then carry out the above coordinate transformation, turning to zero the connection coefficients. This frame, in which the metric tensor differs from $\eta_{\mu\nu}=diag(1,-1,-1,-1)$ only in second derivatives, is called the locally inertial frame, or locally geodesic frame. Its existence is very useful for calculations, as will be seen in the problem on Bianchi identity