Yes, roots need oxygen just like other organisms. When it's not available, the roots will die. Beneficial soil microorganisms also need oxygen to mineralize nutrients in organic matter and to fix nitrogen for legumes such as beans and peas. One group of anaerobic bacteria, the denitrifiers, actually take oxygen from fertilizer nitrates and convert the nitrogen back into a gas that is lost from the soil. One of the obvious consequences of a wet soil is a cold soil in early spring.
More heat is required to warm water than soil.
Therefore, if the soil is well drained, it will warm up rapidly in early spring, allowing earlier planting, fewer seedling diseases, and, perhaps, a longer growing season. Add organic material like compost to open a soil up, allowing more air space and improving drainage. The raised beds also warm up faster in the spring. In areas of poorly drained clay soils, build up additional topsoil or a soil mix on the existing soil surface. Avoid digging deeply into the existing soil, because this could create a bathtub effect and actually hold water.