'that' refers to the thing that he encountered, namely, the incredibly fertile 8,500-square-mile triangle between Cairo and the Mediterranean coast. He could have used 'an' instead of 'that', but using 'that' suggests that the thing he encountered is already known to the reader. I encountered a wide, slow flowing river - just a river, that I encountered, that I wouldn't expect the listener/reader to know about. I encountered THAT wide, slow flowing river - (e.g. the Nile) suggests that the river in question is famous an widely known and the listener/reader has certainly heard of it already.

There is no need for a verb. The entire noun phrase ' that incredibly fertile 8,500-square-mile triangle between Cairo and the Mediterranean coast' refers to the Nile Delta.