POSIX stands for Portable Operating System Interface. It's a set of standards that define a common interface for operating systems. Unix was the original operating system developed but POSIX came around as a standardization to address issues between Unix variants.
In today's day and age, most modern Unix implementations are POSIX compliant. How does this help? POSIX defines a set of APIs along with command line shells and utility interfaces. This means that applications can be built to run on any POSIX-compliant os. This introduces portability & consistency and allows developers to write applications to be run on a wide variety of systems while also allowing users to run their applications on the os of their choice.
POSIX kernels are kernels that are compliant with the POSIX standards. Hence, a POSIX-compliant kernel provides a set of POSIX APIs and other features that allow software to be developed and run in a portable manner on that kernel.
Many Unix-based operating systems, such as Linux, macOS, and most versions of Unix, are based on POSIX-compliant kernels, and they provide a wide range of POSIX APIs and other features that allow software to be developed and run in a portable manner on these systems. This means that software developed using POSIX APIs and standards can be easily compiled and run on these systems, without needing extra modification.
Translation
Now we can talk about the translation bit. Kubernetes is an open-source container orchestration system that is used to automate the deployment, scaling, and management of containerized applications. Kubernetes is built on multiple technologies including the Linux operating system. Since it is built on top of Linux, it takes advantage of the various APIs and features provided by the Linux kernel. This includes the POSIX APIs among other features. Hence as Kubernetes does use these APIs, it means that software built with keeping the POSIX standard in mind runs easily on a kubernetes cluster.
Overall, while POSIX and its surrounding APIs and features are an important part of the underlying technology stack that powers Kubernetes, they are not directly related to the Kubernetes orchestration system itself but since Kubernetes is built on top of the Linux os, it means that Kubernetes uses the APIs provided by the Linux kernel which in turn means using some, if not all POSIX APIs.