Answer:
The phrase shows that the duke is disgusted by his wife's admirers and thinks that they consider themselves more valuable than what they really are.
Explanation:
The word "officious" indicates that someone considers himself too important and is full of himself, but that he doesn't really have that much importance. When the duke uses this word to refer to his wife's admirers, he uses the word in a derogatory way, expressing disgust at the behaviors he is observing both for his wife and for the admirers she presents.
