Leguminous plants like Monantha vetch, hairy vetch and Austrian peas have nitrogen fixing bacteria in their root nodules. Plants cannot consume the atmospheric nitrogen directly. It has to be fixed and made it available to the soil for them to absorb it. This is done by the nitrogen fixing bacteria like rhizobium present in the root nodules of legumes.
Growing legumes on a field adds nitrogen to it. Growing any other crop (in this case-corn) after legumes, benefits that crop as it can absorb the available nitrogen in the soil. Grasses like rye cannot fix nitrogen, rather they consume the available nutrients and the next crop will have less of those nutrients. Leaving the land barren also does not help the crop to be grown on it as no new nutrients are added to the soil before that crop is grown.