The purpose of this operator is to allow the retrieval of documents that match, for example, two predicates A and B that apply to different kinds of document items.
If replaced by an AND then the document would not be retrieved as no document item matches both A and B. If replaced by an OR then documents containing only items matching A (or B) would be retrieved too.
In the output feed, the document items for a given document that matches the WITH predicate will be the union of all document items matching A or B.
For example, refer to Doc Items Filtering Examples.