That person who is both an editor and a publisher is my cousin.
Literal
Editor-and-publisher is that person [topic-は] my cousin is.
兼 is a compact connector meaning 'concurrently/also serving as' — 編集者兼出版者 is 'editor-cum-publisher.' The である relative clause modifies その人: 'the person who is an editor-and-publisher.' This kind of structural density is typical of written Japanese.