Bill O'Reilly on John Edwards

During the O'Reilly Factor, Bill O'Reilly said the following:

O’Reilly: As for John Edwards, Good grief! this guy has no clue. (plays clip)

Edwards: … and tonight, 200,000 men and women who wore our uniform proudly and served this country courageously as veterans will go to sleep under bridges and on grates. We are better than this. (see Edward’s speech here)

O’Reilly: That was Edwards’ concession speech last night. I mean, come on. The only thing sleeping under a bridge is that guy’s brain. 10 million illegal alien workers are sending billions of dollars back home and Edwards is running around saying nobody has any money. Hard to believe.

Why is O'Reilly making fun of this specific point, and what do "illegal alien workers" have to do with it? When O'Reilly says that "[t]he only thing sleeping under a bridge is that guy’s brain," it certainly appears that he is challenging the factual accuracy of Edwards's claim. But there really are 195,827 homeless veterans, and there is certainly something wrong with that.

UPDATE: The Washington Post has more here:

Several readers have asked us to check this surprising statistic, often used by Edwards. The language may be overly dramatic, but the figure is an official one, from the Department of Veterans Affairs. The department believes that one-third of the adult homeless population of the United States "have served their country in the Armed Services." A posting on the department Web site says that about 195,000 veterans are "homeless on any given night" and perhaps twice as many experience homelessness at some point during the course of a year.