You get people hanging towels over the railings, you get clotheslines in the backyard. We just don't like the look of it. It looks like a lower-class neighborhood.
I get my washing done by the nearby laundry, so I don't really need a clothes line, but I'm guessing that part of the subtext here is "I don't want people looking at my knickers" or, perhaps, "I don't want to look at other peoples knickers". As the Right To Dry Campaign has it:
Where in Victorian times, clotheslines were ubiquitous, Mrs. Brown's brassiere blowing in the breeze has apparently become scandalizing to some modern Americans. A strange brand of prudery has made it impossible for some people to conserve energy and money by using a clothesline.
Thanks to Fred over at Slacktivist for this, he goes on to explain about Homeowners Associations:
[W]hy on earth anyone would voluntarily submit to live in such prefab neighborhoods where, it seems, all that is not expressly permitted is forbidden. One could argue that this intrusive corporate governance of private life is un-American. But then I suppose one could also argue that the voluntary surrender of personal freedom in the hopes of attaining higher "property values" is quintessentially American.