For a real vector space $V$ with dimension $n$ and a basis $\{e_1,\dots,e_n\}$ the complexification $V^\mathbb{C}=V \otimes \mathbb C$ contains of vectors of the form $v+iw$ for $v,w\in V$. Now write $v=v^je_j$ and $w=w^je_j$ $$v+iw=v^je_j+iw^je_j=(v^j+iw^j)e_j$$ the basis $\{e_1,\dots,e_n\}$ forms a basis for $V^\mathbb{C}$ also.
Now in with complex manifolds if we have local coordinates $(z_1,\dots,z_n)$ at $p \in M$ which we identify with $(x_1,\dots,x_n,y_1,\dots,y_n)\in \mathbb{R}^{2n}$ the tangent space for $T_pM$ is $$\left\{ \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial y_1}, \dots, \frac{\partial}{\partial y_n} \right\}.$$
So as in the previous case with $V$ and $V^\mathbb{C}$ this should also serve as a basis for $T_pM^{\mathbb{C}}$.
I'm seeing that a basis for $T_pM^\mathbb{C}$ is given by $$\left\{ \frac{\partial}{\partial z_1}, \dots, \frac{\partial}{\partial z_n}, \frac{\partial}{\partial \bar{z}_1}, \dots, \frac{\partial}{\partial \bar{z}_n} \right\}$$ where $$\frac{\partial}{\partial z_j} = \frac{1}{2}\left( \frac{\partial}{\partial x_j} - i \frac{\partial}{\partial y_j} \right) \ \text{ and } \ \frac{\partial}{\partial \bar{z}_j} = \frac{1}{2}\left( \frac{\partial}{\partial x_j} + i \frac{\partial}{\partial y_j} \right).$$
Why is this a basis and why are $\frac{\partial}{\partial z_j}$ and $\frac{\partial}{\partial \bar{z}_j}$ defined as they are?
We have $$ \left( \begin{array}{c} \partial_{z_1}\\ \vdots \\ \partial_{z_n}\\ \partial_{\overline{z_1}}\\ \vdots\\ \partial_{\overline{z_n}} \end{array}\right) =G\left( \begin{array}{c} \partial_{x_1}\\ \vdots\\ \partial_{x_n}\\ \partial_{y_1}\\ \vdots\\ \partial_{y_n} \end{array} \right) $$ where $G=\frac 12\left(\begin{array}{c|c} I & -I \\ \hline I & I \end{array}\right)$ is an invertible matrix. (Here $I$ is the $n\times n$ identity matrix.)
Therefore, if you know that $\lbrace\partial_{x_1},\dots,\partial_{y_n}\rbrace$ is a basis (over $\mathbb C$) of $T_pM^\mathbb C$, also $\lbrace\partial_{z_1},\dots,\partial_{\overline{z_n}}\rbrace$ must be a basis (over $\mathbb C$) of $T_pM^\mathbb C$.
Why are they defined in this way? This boils down to why (in Complex Analysis) the Wirtinger operators are defined in this way and they are useful (https://en.wikipedia.org/wiki/Wirtinger_derivatives). Essentially, they encode complex differentiation and the Cauchy-Riemann equations. For example, holomorphic functions are the kernel of $\partial_{\overline {z_i}}$. In Complex Geometry, similarly, they allow us to define holomorphic functions, holomorphic tangent spaces, holomorphic forms,... (see https://en.wikipedia.org/wiki/Holomorphic_tangent_bundle)
Moreover, they serve as a basis in which the complex structure $J$ $$ J\partial_{x_s}=\partial_{y_s},\quad J\partial_{y_s}=-\partial_{x_s}\qquad (s=1,\dots,n) $$ is diagonal $$ J\partial_{z_s}=i\partial_{z_s},\quad J\partial_{\overline{z_s}}=-i\partial_{\overline{z_s}} \qquad(s=1,\dots,n)\,. $$ (See https://en.wikipedia.org/wiki/Almost_complex_manifold for more information.)