The positive impacts of illegal immigration are similar to the positive effects of immigration in general. They are an increased work force and lower costs for consumers.
With immigration, the size of the US labor force increases. This may sound bad at a time when we have high unemployment, but it is good in the long run because it helps our economy to keep going. Countries like Japan that have little immigration are facing declining workforces, which makes it hard (among other things) to support the people who are retired or will soon retire. In the US right now, we need young workers to pay social security taxes and help keep Social Security solvent as the baby boomers retire.
With immigration, we also get lower costs for consumers. Immigrants are typically willing to do hard and unpleasant jobs for relatively low wages. Because immigrants are willing (for example) to work for low wages in meat processing plants, we get cheap food.