H. pylori helps regulate stomach acidity, the byproduct of which is sometimes ulcers. But when it is taken out of the picture, stomach acidity can increase and cause esophageal reflux disease, a disease which has increased to match the decrease in ulcers as H. pylori has been eradicated.
The asthma mechanism is less clear, but may be from H. pylori stimulating immune response. The evidence supporting the link is that U.S. children aged 3-13 who have H. pylori are 60% less likely to have asthma than those who do not.
The obesity connection is also not definitively established, but people without H. pylori produce more grehlin (which makes you feel hungry) than those who have it.
(Via "The twists and turns of fate," about the work of Martin Blaser, a microbiologist at New York University School of Medicine, in The Economist, August 23, 2008, pp. 68-69.)