In the story "A Rose for Emily" it says "We were glad Miss Emily was getting married because the two female cousins were even more Grierson.........

One important theme of this story is the social class
difference between the elite of the town, like Emily Gierson and her family, in
comparison to pretty much everyone else.  Miss Emily comes from an upper class family
and lives on "what was once our most august street."  She is also of the older
generation where class distinctions where more strict and more respected.  Miss Emily is
treated differently than many other people in the town may have been treated just
because of who she is and who her father was.  For this reason, the town council remit
her taxes after her father dies, but the next generation of aldermen who don't put as
much stock in social classes try to get her to pay those taxes.  She still lives in her
own world where her name means something, and she vanquishes them from her home and
never does acknowledge the taxes bills that arrive each year.  The respect and deference
that she commands comes from her understanding of who she is in this society -- she is a
Grierson!  That is why the druggist gives her the poison without her stating why she
needs it.  That is way the men sprinkle the lime around her house without ever
determining the cause of the stench.  That is why they don't insist she have postal
numbers put up on her house. 

The Grierson cousins are
probably of the same arrogant attitude.  There are no details in the story, but from the
line you quoted, it is to be inferred that they are elitist snobs who think and act as
if they are better than everyone else.  If Miss Emily acts in a way that intimidates
those around her, the cousins probably do as well.  It sounds like they may actually be
worse!  The townspeople are not big supporters of Miss Emily.  They think she is odd,
aloof, and arrogant, but the narrator tells us that the townspeople were kind of in Miss
Emily's corner here.  Miss Emily didn't want her relatives there to "keep an eye on her"
in regards to her relationship with Homer, so now that they appear to be acting
responsibly and getting married, everyone -- the townspeople and Miss Emily -- will be
happily rid of the cousins who will no longer have a purpose to be in town.  The
cousins' presence in the story helps to illustrate the theme of the class distinctions
in the South at the turn of the 20th century.