Why does the doubly linked list in sys/queue.h maintain the address of previous next element?

If you would have read the queue.h file from the beginning, you may have got following comment:

 * A list is headed by a single forward pointer (or an array of forward
 * pointers for a hash table header). The elements are doubly linked
 * so that an arbitrary element can be removed without a need to
 * traverse the list. New elements can be added to the list before
 * or after an existing element or at the head of the list. A list
 * may only be traversed in the forward direction.

so list, which provides O(1) insertion and deletion, but only forward traversal. To achieve this, you only need the reference to the previously next pointer, which is exactly what is implemented.