So far, nobody has a definitive answer about how Viagra and other drugs in the PDE-5 inhibitor class (such as Levitra and Cialis) help with premature ejaculation, but there are several theories. Some researchers posit that Viagra acts on parts of the brain that control ejaculation. Apparently, too much nitric oxide in the brain inhibits ejaculation, and Viagra (and drugs like it) increase the amount of nitric oxide in the body. Nobody definitively knows that Viagra increases nitric oxide in the brain, but there is preliminary scientific evidence that it does.
Other researchers theorize that Viagra has a psychological effect on its user, the logic being that a firmer erection makes a man feel more in-control, and that alone can help forestall premature ejaculation.
Another theory is purely practical: Viagra and drugs like it may not prevent premature ejaculation first time around, but they make it easier to get a second erection, which is more likely to last longer than the first one.