You need to differentiate the noun from the verb, then investigate the verb for your answer.
The noun doesn't help much here.
root n.: the part of a plant, usually below the ground, that lacks nodes, shoots, and leaves, holds the plant in position, draws water and nourishment from the soil, and stores food
Origin: Middle English rote from Late Old English from Old Norse rot, akin to Old English wyrt, German wurzel from Indo-European base an unverified form wrād-, twig, root from source Glassical Greek rhiza, Classical Latin radix, root, ramus, branch
root vi.: to give audible encouragement or applause to a contestant or team; cheer. See Synonyms at applaud; to lend support to someone or something.
Origin: possibly alteration of rout. A second source also mentioned rout as the possible origin. root:Possibly an alteration of rout (“to make a loud noise”), influenced by hoot
rout to bellow, used of cattle. (First Known Use: 14th century): 14th century (Middle English rowten, from Old Norse rauta; akin to Old English rēotan to weep, Latin rudere to roar)