By ending the sentence like this, the speaker is implying he has something more to say. His wife died, his kids left home, and that may not be the end of his story. Or he may just want to add how sad he was.
He may continue his story right after this sentence, but the remaining part may be simply omitted when it's obvious.
"Let's go to a movie tomorrow." "Er, I don't have money now, so..."
The で in 死んで is basically the same as て, but て will be voiced after some verbs (rendaku phenomenon).
(Not to be confused て as a casual request.)