Roots need oxygen for cellular respiration. The above ground part of the plant can get oxygen from the atmosphere (the leaves are actually trying to get rid of excess oxygen) and through openings in the stems called lenticels.  Below ground is a mother matter. When you see paths worn in the grass where people walk consistently, the plants have usually died not from being walked on, but from soil compaction which eliminates air spaces in the soil as well as making the soil difficult for the roots to pass through.