I suggest learning the basics of conformal field theory before diving into string theory. To illustrate where holomorphicity comes in, let's consider the free bosonic string (the simplest example of a CFT). One starts with the worldsheet embedding $\phi(z,\overline{z}): \mathbb{C}P^1 \to \mathbb{C}$ and derives the equations of motion $\partial_{\overline{z}} \partial_z \phi(z,\overline{z}) = 0$. This implies the existence of the fields
\begin{equation}
\partial \phi(z) = \sum_{n \in \mathbb{Z}} a_n z^{-n-1} \ \ \ \ \ \ \ \ \ \ \overline{\partial} \phi(\overline{z}) = \sum_{n \in \mathbb{Z}} \overline{a}_n \overline{z}^{-n-1},
\end{equation}
which are holomorphic and antiholomorphic respectively. So one answer to your question is that the fields of interest are holomorphic/antiholomorphic as a result of the equations of motion (which in the case of the free boson are derived from the action of the theory).
There is much more to be said here. To obtain a quantum theory, the coefficients $a_n$ are promoted to elements of a Lie algebra. This is known as canonical quantisation. In the case of the free boson, the Lie algebra is generated by the elements $\{a_n\}_{n \in \mathbb{Z}}$ and $\{k\}$, with Lie bracket defined by
\begin{equation}
[a_n, a_m] = \delta_{n, -m}k \ \ \ \ \ [a_n, k] = 0.
\end{equation}
There is another Lie algebra of interest known as the Virasoro algebra. The Virasoro algebra appears in 2d CFT as a result of conformal invariance of the theory. The Virasoro algebra has generators $L_n$ and one obtains another holomorphic field
\begin{equation}
T(z) = \sum_{n \in \mathbb{Z}} L_n z^{-n-2}
\end{equation}
known as the energy-momentum tensor. There is also an antiholomorphic analogue of the energy-momentum tensor. For this reason, we say the free boson is composed of holomorphic and antiholomorphic "sectors".
If you are interested in learning more about CFT, check out Conformal Field Theory by Sénéchal, Di Francesco, and Mathieu. For a more mathematically rigorous account, see Vertex Algebras and Algebraic Curves by Frenkel and Ben-Zvi (your background in algebraic geometry will be useful here).
