Abstract
In $\boldsymbol{\S \rm A }$ we give some notes from the theory of 2d-surfaces in $\mathbb{R}^3$ useful for the interpretation of the corresponding theory in higher dimensions. In $\boldsymbol{\S \rm B}$ we make the connection with the question and in $\boldsymbol{\S \rm C}$ we give a $n$-dimensional generalization.
$\boldsymbol{\S \rm A. }$ FROM THE THEORY OF 2D-SURFACES
A real 3-vector function of two real parameters $(u,v)$
\begin{equation}
\mathbf{x}\left(u,v\right)\boldsymbol{=}f_1\left(u,v\right)\mathbf{e}_1\boldsymbol{+}f_2\left(u,v\right)\mathbf{e}_2\boldsymbol{+}f_3\left(u,v\right)\mathbf{e}_3
\tag{01}\label{01}
\end{equation}
where $\left(\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\right)$ is a basis in $\mathbb{R}^3$, represents a two-dimensional surface. Its partial derivatives are denoted by
\begin{align}
\mathbf{x}_{u} & \boldsymbol{=}\dfrac{\partial \mathbf{x}}{\partial u}\,,\quad \mathbf{x}_{uu}\boldsymbol{=}\dfrac{\partial}{\partial u}\left(\dfrac{\partial \mathbf{x}}{\partial u}\right)\boldsymbol{=}\dfrac{\partial^2 \mathbf{x}}{\partial u^2}\,,\quad \mathbf{x}_{uv}\boldsymbol{=}\dfrac{\partial}{\partial v}\left(\dfrac{\partial \mathbf{x}}{\partial u}\right)\boldsymbol{=}\dfrac{\partial^2 \mathbf{x}}{\partial v\partial u}\quad \texttt{etc}
\nonumber\\
\mathbf{x}_{v} & \boldsymbol{=}\dfrac{\partial \mathbf{x}}{\partial v}\,,\quad \mathbf{x}_{vv}\boldsymbol{=}\dfrac{\partial}{\partial v}\left(\dfrac{\partial \mathbf{x}}{\partial v}\right)\boldsymbol{=}\dfrac{\partial^2 \mathbf{x}}{\partial v^2}\,,\quad \mathbf{x}_{vu}\boldsymbol{=}\dfrac{\partial}{\partial u}\left(\dfrac{\partial \mathbf{x}}{\partial v}\right)\boldsymbol{=}\dfrac{\partial^2 \mathbf{x}}{\partial u\partial v}\quad \texttt{etc}
\tag{02}\label{02}
\end{align}
At a point $\mathrm P$ on the surface with curvilinear coordinates $\left(u,v\right)$, see Figure-01, the vectors
\begin{equation}
\mathbf{x}_{u} \boldsymbol{\equiv}\dfrac{\partial \mathbf{x}}{\partial u}\qquad \mathbf{x}_{v} \boldsymbol{\equiv}\dfrac{\partial \mathbf{x}}{\partial v}
\tag{03}\label{03}
\end{equation}
are a basis with respect to the curvilinear coordinates $\left(u,v\right)$. The vector $\mathbf{x}_{u}$ is tangent to the $u\boldsymbol{-}$parametric curve $v\boldsymbol{=}\texttt{constant}$ while the vector $\mathbf{x}_{v}$ is tangent to the $v\boldsymbol{-}$parametric curve $u\boldsymbol{=}\texttt{constant}$. The vectors $\mathbf{x}'_{u},\mathbf{x}'_{v}$ tangent to the parametric curves on a second point $\mathrm P'$ are a basis on this point as shown in the same Figure-01.
The members of a basis $\{\mathbf{x}_{u},\mathbf{x}_{v}\}$ are not in general unit vectors. Also they are not in general perpendicular to each other. In Figure-02 we see two points $\mathrm P\left(u,v\right)$ and $\mathrm P\left(u\boldsymbol{+}\mathrm du,v\boldsymbol{+}\mathrm dv\right)$ on the surface infinitesimally close to each other.
The infinitesimal vector
\begin{equation}
\mathrm d\mathbf{x}\boldsymbol{=} \mathbf{x}_u \mathrm du\boldsymbol{+} \mathbf{x}_v \mathrm dv
\tag{04}\label{04}
\end{equation}
has the property
\begin{equation}
\mathbf{x}\left(u\boldsymbol{+}\mathrm du,v\boldsymbol{+}\mathrm dv\right)\boldsymbol{=}\mathbf{x}\left(u,v\right)\boldsymbol{+}\mathrm d\mathbf{x}\boldsymbol{+}\mathbf{o}\left(\left(\mathrm du^2\boldsymbol{+}\mathrm dv^2\right)^{1/2}\right)
\tag{05}\label{05}
\end{equation}
Thus the vector $\mathrm d\mathbf{x}$ is a first order approximation to the vector $\mathbf{x}\left(u\boldsymbol{+}\mathrm du,v\boldsymbol{+}\mathrm dv\right)\boldsymbol{-} \mathbf{x}\left(u, v\right)$ from the point $\mathrm P\left(u,v\right)$ to the neighboring point $\mathrm P\left(u\boldsymbol{+}\mathrm du,v\boldsymbol{+}\mathrm dv\right)$ on the surface.
For the infinitesimal length $\mathrm ds^2$ we have
\begin{align}
\mathrm ds^2 & \boldsymbol{=}
\mathrm d\mathbf{x}\boldsymbol{\cdot}\mathrm d\mathbf{x}\boldsymbol{=}\left(\mathbf{x}_{u}\mathrm du\boldsymbol{+}\mathbf{x}_{v}\mathrm dv\right)\boldsymbol{\cdot}\left(\mathbf{x}_{u}\mathrm du\boldsymbol{+}\mathbf{x}_{v}\mathrm dv\right)
\nonumber\\
& \boldsymbol{=}\left(\mathbf{x}_{u}\boldsymbol{\cdot}\mathbf{x}_{u}\right)\mathrm du^2\boldsymbol{+}2\left(\mathbf{x}_{u}\boldsymbol{\cdot}\mathbf{x}_{v}\right)\mathrm du \mathrm dv\boldsymbol{+}\left(\mathbf{x}_{v}\boldsymbol{\cdot}\mathbf{x}_{v}\right)\mathrm dv^2
\nonumber\\
&\boldsymbol{=}g_{uu}\mathrm du^2\boldsymbol{+}2g_{uv}\mathrm du \mathrm dv\boldsymbol{+}g_{vv}\mathrm dv^2
\tag{06}\label{06}
\end{align}
where
\begin{align}
g_{uu} & \boldsymbol{=} \left(\mathbf{x}_{u}\boldsymbol{\cdot}\mathbf{x}_{u}\right)\boldsymbol{=}\Vert\mathbf{x}_{u} \Vert^2
\tag{07a}\label{07a}\\
g_{uv} & \boldsymbol{=}\left(\mathbf{x}_{u}\boldsymbol{\cdot}\mathbf{x}_{v}\right)\boldsymbol{=}\left(\mathbf{x}_{v}\boldsymbol{\cdot}\mathbf{x}_{u}\right)\boldsymbol{=}g_{vu}
\tag{07b}\label{07b}\\
g_{vv} & \boldsymbol{=}\left(\mathbf{x}_{v}\boldsymbol{\cdot}\mathbf{x}_{v}\right)\boldsymbol{=}\Vert\mathbf{x}_{v} \Vert^2
\tag{07c}\label{07c}
\end{align}
the elements of the metric tensor
\begin{equation}
g_{ij} \boldsymbol{=}
\begin{bmatrix}
g_{uu} & g_{uv}\vphantom{\dfrac{a}{b}} \\
g_{vu} & g_{vv}\vphantom{\dfrac{a}{b}}
\end{bmatrix}
\tag{08}\label{08}
\end{equation}
In Figure-03 we have a vector $\boldsymbol{\xi}$ in the plane tangent to the point
$\mathrm P\left(u,v\right)$ expressed in terms of the basis $\{\mathbf{x}_{u},\mathbf{x}_{v}\}$
\begin{equation}
\boldsymbol{\xi} \boldsymbol{=} \xi_u\mathbf{x}_{u}\boldsymbol{+}\xi_v\mathbf{x}_{v}
\tag{09}\label{09}
\end{equation}
So $\boldsymbol{\xi}\boldsymbol{=}\left(\xi_u,\xi_v\right)$ is the representation of this vector with respect to the curvilinear system of coordinates $\left(u,v\right)$ with magnitude squared
\begin{equation}
\Vert\boldsymbol{\xi}\Vert^2 \boldsymbol{=} g_{uu}\vert\xi_u\vert^2\boldsymbol{+}2g_{uv}\xi_u \xi_v\boldsymbol{+}g_{vv}\vert\xi_v\vert^2
\tag{10}\label{10}
\end{equation}
Suppose now that $\mathbf{x}\boldsymbol{=}\mathbf{x}^{\boldsymbol{*}}\left(\theta,\phi\right)$ is an other parametric representation of the surface, that is an other system of curvilinear coordinates
\begin{equation}
\theta \boldsymbol{=}\theta \left(u,v\right)\,,\qquad \phi \boldsymbol{=}\phi \left(u,v\right)
\tag{11}\label{11}
\end{equation}
At a point $\mathrm P$ on the surface having curvilinear coordinates $\left(u,v\right)$ and $\left(\theta,\phi\right)$ in the two systems respectively we have for the tangent vectors of latter system
\begin{align}
\mathbf{x}^{\boldsymbol{*}}_{\theta} & \boldsymbol{=}
\dfrac{\partial \mathbf{x}^{\boldsymbol{*}}}{\partial \theta}\boldsymbol{=}\dfrac{\partial \mathbf{x}}{\partial u}\dfrac{\partial u}{\partial \theta} \boldsymbol{+}\dfrac{\partial \mathbf{x}}{\partial v}\dfrac{\partial v}{\partial \theta}\boldsymbol{=}u_\theta\mathbf{x}_u \boldsymbol{+} v_\theta\mathbf{x}_v
\tag{12a}\label{12a}\\
\mathbf{x}^{\boldsymbol{*}}_{\phi} & \boldsymbol{=}
\dfrac{\partial \mathbf{x}^{\boldsymbol{*}}}{\partial \phi}\boldsymbol{=}\dfrac{\partial \mathbf{x}}{\partial u}\dfrac{\partial u}{\partial \phi} \boldsymbol{+}\dfrac{\partial \mathbf{x}}{\partial v}\dfrac{\partial v}{\partial \phi}\boldsymbol{=}u_\phi\mathbf{x}_u \boldsymbol{+} v_\phi\mathbf{x}_v
\tag{12b}\label{12b}
\end{align}
as shown in Figure-04. So
\begin{equation}
\begin{bmatrix}
\mathbf{x}^{\boldsymbol{*}}_{\theta} \vphantom{\dfrac{\tfrac{a}{b}}{\dfrac{a}{b}}}\\
\mathbf{x}^{\boldsymbol{*}}_{\phi} \vphantom{\dfrac{a}{\tfrac{a}{b}}}
\end{bmatrix}
\boldsymbol{=}
\begin{bmatrix}
\:\:\dfrac{\partial u}{\partial \theta} \:& \:\dfrac{\partial v}{\partial \theta}\:\:\vphantom{\dfrac{a}{\dfrac{a}{b}}} \\
\:\:\dfrac{\partial u}{\partial \phi} \:& \:\dfrac{\partial v}{\partial \phi}\:\:\vphantom{\dfrac{a}{\tfrac{a}{b}}}
\end{bmatrix}
\begin{bmatrix}
\mathbf{x}_{u} \vphantom{\dfrac{\tfrac{a}{b}}{\dfrac{a}{b}}} \\
\mathbf{x}_{v} \vphantom{\dfrac{a}{\tfrac{a}{b}}}
\end{bmatrix}
\tag{13}\label{13}
\end{equation}
Note that $\{\mathbf{x}^{\boldsymbol{*}}_{\theta},\mathbf{x}^{\boldsymbol{*}}_{\phi}\}$ is a basis at point $\mathrm P$ with respect to the new system of curvilinear coordinates $\left(\theta,\phi\right)$.
All vectors $\mathbf{x}_{u},\mathbf{x}_{v},\mathbf{x}^{\boldsymbol{*}}_{\theta},\mathbf{x}^{\boldsymbol{*}}_{\phi}$ are coplanar : they belong to the plane tangent to the point $\mathrm P$.
$\boldsymbol{\S \rm B. }$ CONNECTION WITH THE QUESTION
\begin{equation}
\vec{e}_{\alpha'}\boldsymbol{=}\Lambda^\beta_{~\alpha'}\vec{e}_{\beta}\boldsymbol{=}\frac{\partial x^\beta}{\partial x^{\alpha'}}\vec{e}_{\beta}
\tag{1 of the question}\label{1 of the question}
\end{equation}
To interpret above equation \eqref{1 of the question} for the two-dimensional case let make the following correspondences
\begin{align}
& u \boldsymbol{=} x^1\qquad v\boldsymbol{=} x^2\qquad \theta\boldsymbol{=} x^{1'}\qquad \phi\boldsymbol{=} x^{2'}
\tag{14a}\label{14a}\\
&\mathbf{x}_{u}\boldsymbol{=}\vec{e}_{1}\qquad\mathbf{x}_{v}\boldsymbol{=}\vec{e}_{2} \qquad\mathbf{x}^{\boldsymbol{*}}_{\theta}\boldsymbol{=}\vec{e}_{1'}\qquad\mathbf{x}^{\boldsymbol{*}}_{\phi}\boldsymbol{=}\vec{e}_{2'}
\tag{14b}\label{14b}\\
&\dfrac{\partial u}{\partial \theta}\boldsymbol{=}\frac{\partial x^1}{\partial x^{1'}}\qquad\dfrac{\partial v}{\partial \theta}\boldsymbol{=}\frac{\partial x^2}{\partial x^{1'}} \qquad\dfrac{\partial u}{\partial \phi}\boldsymbol{=}\frac{\partial x^1}{\partial x^{2'}}\qquad\dfrac{\partial v}{\partial \phi}\boldsymbol{=}\frac{\partial x^2}{\partial x^{2'}}
\tag{14c}\label{14c}
\end{align}
Then equation \eqref{13} corresponds to the following
\begin{equation}
\vec{e}_{\alpha'}\boldsymbol{=}\frac{\partial x^\beta}{\partial x^{\alpha'}}\vec{e}_{\beta} \qquad \alpha'\boldsymbol{=}1',2' \qquad \beta\boldsymbol{=}1,2
\tag{15}\label{15}
\end{equation}
$\boldsymbol{\S \rm C. }$ N-DIMENSIONAL GENERALIZATION
Consider that $\mathbf{x}\boldsymbol{=}\mathbf{x}\left(x^1,x^2,\cdots,x^\beta,\cdots,x^n\right)\in \mathbb R^{n+1}$ is a real $(n+1)$-vector where $x^\beta\in \mathbb R$ are $n$ real parameters. Then $\mathbf{x}$ is a parametric representation of a $n$-dimensional $^{\prime\prime}$surface$^{\prime\prime}$(manifold) in $\mathbb R^{n+1}$.
At a point $\mathrm P$ we define the following basis of $n$-vectors
\begin{equation}
\vec{e}_{\beta}\boldsymbol{=}\dfrac{\partial \mathbf{x}}{\partial x^{\beta}}\qquad \beta\boldsymbol{=}1,2,3,\cdots,n
\tag{16}\label{16}
\end{equation}
The vector $\vec{e}_{\beta}$ is tangent to the $x^{\beta}$-parametric curve with $x^{\rho}\boldsymbol{=}\texttt{constant},\rho\boldsymbol{\ne}\beta $. The linear $n$-dimensional space generated by this basis $\{\vec{e}_{1},\vec{e}_{2},\cdots,\vec{e}_{n}\}$ is the tangent space to the point $\mathrm P$.
Under a parametric transformation
\begin{equation}
x^{\alpha'}\boldsymbol{=}x^{\alpha'}\left(x^1,x^2,\cdots,x^\beta,\cdots,x^n\right)\qquad \alpha'\boldsymbol{=}1',2',3',\cdots,n'
\tag{17}\label{17}
\end{equation}
we have a new parametric representation
\begin{equation}
\mathbf{x}'\left(x^{1'},x^{2'},\cdots,x^{\alpha'},\cdots,x{n'}\right)\boldsymbol{=}\mathbf{x}\left(x^1,x^2,\cdots,x^\beta,\cdots,x^n\right)
\tag{18}\label{18}
\end{equation}
and so a new basis of tangent $n$-vectors
\begin{equation}
\vec{e}_{\alpha'}\boldsymbol{=}\dfrac{\partial \mathbf{x}'}{\partial x^{\alpha'}}\boldsymbol{=}\dfrac{\partial \mathbf{x}}{\partial x^{\beta}}\dfrac{\partial x^\beta}{\partial x^{\alpha'}}\boldsymbol{=}\dfrac{\partial x^\beta}{\partial x^{\alpha'}}\vec{e}_{\beta}\: \left(\alpha'\boldsymbol{=}1',2',3',\cdots,n'\:\Vert\:\beta\boldsymbol{=}1,2,3,\cdots,n\right)
\tag{19}\label{19}
\end{equation}