If your neighbor is correct, then it's likely you have sod webworms (which do, in fact, use silk to create their tiny little burrows in your lawn). There are numerous products you can purchase at garden centers for control. Bacillus thuringiensis ("BT") is a "biorational" option, there are several synthetic options, and the really organic (and also fairly expensive) option is parasitic nematodes, which you can obtain by mail order, or through retail garden centers that specialize in organics. They all work pretty well if used according to package directions.
Having said that, you will want to keep a close eye on your grass to determine whether this is really the problem. After all, lots of things can turn your St. Augustinegrass brown. One of them is chinch bugs. Chinch bugs are small, black insects, and when they're numerous enough to turn your lawn brown, you shouldn't have any trouble seeing them crawling all over the place. Webworms inhabit tiny silken burrows, which you can spot with close inspection.
Whatever the problem, keep your lawn healthy and well watered in summer, and also dethatch your lawn annually, if you don't do so already. These two steps might reduce the problem to a tolerable level that doesn't require you to spray anything at all.