The couple next door are fighting again — one of those spats dogs wouldn't even touch.
Literal
Next [at-に] [topic-は] again dogs even won't-eat marital-fight [object-を] is-doing [informative-よ].
犬も食わない夫婦喧嘩 ('a marital fight even dogs won't touch') is a classic proverb-based idiom about minor domestic squabbles. The underlying proverb 夫婦喧嘩は犬も食わない reflects the idea that petty couple fights are so pointless even dogs (which eat anything) wouldn't bother with them. 隣 alone (without 家) works as 'next door.'