I am not sure how much they increase the oxygen, but I only bought two plants a few weeks ago and they've multiplied like crazy in my ten gallon. My nitrates were only 10ppm to begin with, they're down to 5ppm now. You can go find the "Proven Performers" in the planted tanks section of this forum to find out which plants do well with minimal care. SOme are java fern, java moss. I have Hygrophilia polyspermum and love it. If your tank is established, the plants will really eat up your nitrates and grow quickly. I tripled my plant volume in just a few weeks.
I have no idea how much they increase oxygen in comparison to an airstone, though. It's one or the other and I think the plants will be more expensive. But with the plants you get the added benefit of reducing your nitrates.