Geometry, Dynamics And Topology Of Foliations: A First Course, Book by Bruno Scárdua and Carlos Arnoldo Morales Rojas, Chapter 1, Page 33. Definition 1.14.
The definition of Holomorphic Foliation on a Complex Manifold.
What do they mean by (Given any intersection $U_i\cap U_j\neq\phi$ the change of coordinates $\varphi_j \circ \varphi_i ^{-1}$ preserves the horizontal fibration on $\mathbb{C}^n \simeq\mathbb{C}^k \times \mathbb{C}^{n-k}$)?
It means that for all $w \in \mathbb C^{n-k}$ there exists $z \in \mathbb C^{n-k}$ such that $$\phi_j \circ \phi_i^{-1}\bigl(\phi_i(U_i \cap U_j) \cap (\mathbb C^k \times \{w\})\bigr) \subset \phi_j(U_i \cap U_j) \cap (\mathbb C^k \times\{z\}) $$ This has nothing to do with holomorphicity or complex manifolds per se. You'll find similar definitions of foliations on real manifolds, with any level of smoothness.