Expressed in local coordinates it becomes an actual function, satisfying certain transformation properties.
Theta functions are naturally thought of as being the states in the geometric quantization of the given complex space, the given holomorphic line bundle being the prequantum line bundle and the condition of holomorphicity of the section being the polarization condition. See for instace (Tyurin).
Introductions to the traditional notion include
D.H. Bailey et al, The Miracle of Theta Functions (web)
M. Bertola, Riemann surfaces and theta functions (pdf)
A modern textbook account is
Further discussion with an emphasis of the origin of theta functions in geometric quantization is in
Arnaud Beuville, Theta functions, old and new (pdf)
Yuichi Nohara, Independence of polarization in geometric quantization (pdf)
Relation to conformal blocks: