AFAIK "spawns" means "creates" with the added implication that every new process is the child of a previous process (except for the first!).
The way that a process creates a new process is via the fork (or vfork) system call. On receiving the fork call, the kernel creates a new process that is an exact duplicate of the parent except for the PID and the return value from the fork call (there may be other differences I have forgotten). Both processes (parent and child) examine the return form the fork call to identify which one they are and then ... whatever the programmer decided -- typically the child executes a system call from the execve family.