The "Create Child" button makes a new card in the stack that is the next level down in the hierarchy (as per field "Child Stack Name"). The Self Name, Self ID, and Self Stack Name (not in any field) are taken from the current card and are set up on the new Child Card as Parent Name, Parent ID, and Parent Stack Name.
The new Child Card gets its Child Stack Name from the 1st card in the new child's stack.
The Parent Card of the new Child Card is told about its new child as the new card's ID is added to the end of the Parent Card's "Child IDs" field.
The cursor is placed in the "Self Name" field to encourage immediate naming by the user.