Eggcorn Forum

I’d always associated it with the way pigs root for acorns, pushing hard against the soil (= pushing for your team). But it seems the etymology is unclear. There’s various suggestions discussed here:

American Heritage Dictionary says “Perhaps from ‘root,’ to dig with the snout.”
Merriam Webster says it may come from “rout.”
Main Entry: 4root
Pronunciation: ‘rüt also ‘rut
Function: intransitive verb
Etymology: perhaps alteration of 2rout
Date: 1889
1 : to noisily applaud or encourage a contestant or team : CHEER
2 : to wish the success of or lend support to someone or something – root.er noun
“The Barnhart Concise Dictionary of Etymology” by Robert K. Barnhart (HarperCollins Publishers, New York, 1995) says it may come from root in the sense of working or study hard. 1856.

Word Detective has this interesting theory:

Meanwhile, the generally accepted theory explains “root,” which first appeared in the sports fan sense in the late 1880s, as an extension of “root” as plants “take root,” i.e., that the fans have bonded closely with, and “sunk their roots into” the team. A more vivid (and to me more plausible) explanation, however, has been suggested by etymologist Gerald Cohen, who points to the foot-stamping, clapping and shouting behavior of baseball fans described by 18th century sportswriters. According to Cohen, foot-stamping (or “pedal-music”) was at that time a vital method of expressing support for one’s team, and “rooting” originally referred to fans stamping their feet so hard that they might actually dig (or “root”) holes in the ground.