The roots of a plant are not often thought of as requiring oxygen; they grow under the ground, where there isn't as much as in the air. However, oxygen is vital to all plants, and it is as important to the roots as it is to the leaves. One way roots get oxygen is through the soil. As James A McAfee, Ph.D. of Texas A&M, says, oxygen for roots "comes from the soil air, which originates from the air above it. Soils must be able to 'breath' in and hold enough oxygen in the soil for roots to grow and function properly."
In addition, roots grow to absorb water and nutrients from the earth underneath the main plant. Water absorbed is pumped up along the plant stem and body by capillaries. Water is H^2O, or two Hydrogen molecules to one Oxygen molecule. Some of this oxygen is split off to help aerate the roots themselves, which will start to rot if over or under-oxygenated. Since both water and oxygen are more available near the surface, the main roots grow feeder roots that spread to an average of three times the surface area of the plant itself. Transfer of oxygen and Carbon Dioxide from the roots is reversed from the leaves; leaves take Carbon Dioxide in and emit Oxygen, while roots take Oxygen in and emit Carbon Dioxide. This is part of Root Respiration, an important part of the life cycle.
If the roots do not receive enough oxygen, root respiration will slow, and so will photosynthesis occurring in the leaves. Without photosynthesis, the plant cannot grow new leaves or bud flowers, and parts of the plant will die. Oxygen also changes the electrical charges in water, and so the roots can absorb faster without drawing on their stores of glucose, which are needed to feed the plant during winter.