The neighbors were fed up because we made a big fuss last night.
Literal
Next [of-の] house [of-の] people [topic-は] we [subject-が] last-night big-fuss [object-を] did [because-ので] were-exasperated.
大騒ぎをする ('to make a fuss, create a ruckus') is the set verb phrase. 閉口する ('to be exasperated, be at a loss, be annoyed') is a Sino-Japanese idiom — literally 'to close the mouth,' i.e., 'can't say anything in response.' Takes no direct object; the cause is expressed with ので.