Green’s Dictionary of Slang

root (for) v.

[ety. unknown; Cohen (ed.), Studies in Slang II (1989), pp.67–8, suggests root = SE dig, and thus an image of cheering and stamping so hard that one ‘digs a hole’ in the grandstand]
(US)

1. to cheer and urge on.

[US]World (N.Y.) 20 Oct. 2/3: In her own feeble way the lone woman under the grand stand ‘rooted’ for the New Yorkers.
[US]Ade Artie (1963) 64: He’ll have the whole push rootin’ for him.
[US]Day Book (Chicago) 22 Oct. 31: We’re rooting for you, Theodore, in sotuh and east and west.
[US]H.C. Witwer Smile A Minute 58: Them roughnecks has started rootin’ for him.
[US]V. Samuels ‘Baseball Sl.’ in AS II:5 256: ‘Pluggers’ or ‘rooters,’ ‘plug’ or ‘root’ for their side or for their favorite players.
[US]J. Weidman What’s In It For Me? 189: You got me rooting for you, Murray.
[US]B. Schulberg Harder They Fall (1971) 52: I was rooting for the coloured boy.
[UK]P. Larkin letter 13 March in Thwaite Sel. Letters (1992) 237: Please don’t think I’m rooting for Jonathan Price to an unreasonable extent, and I shan’t care twopence if he isn’t represented.
[US]G. Marx letter 9 Jan. in Groucho Letters (1967) 119: I was rooting mentally for Max to answer ‘Mikoyan’ to the big-money question.
[US]C. Hiaasen Tourist Season (1987) 320: Michael, who’re you rooting for?
[UK]Indep. on Sun. Real Life 11 July 2: Everyone is rooting for a day trip to Brighton.
[UK]M. Amis Experience 252: I would be rooting for my boy John (Best Actor, Pulp Fiction).

2. to support a cause.

[US]Ade Forty Modern Fables 184: The whole Bunch is going out to Root for Purity in Politics.
[US]S. Lewis Babbitt (1974) 28: Who you rootin’ for for Republican candidate, Mr. Babbitt?
[US](con. 1910s) S. Lewis Elmer Gantry 344: Anybody that roots for him’ll stand ace-high after the election.
[US]E. Gilbert Vice Trap 37: He was trying to get on special with the post office, and I was rooting for him to make it.
[UK]Guardian Guide 15–21 May 93: With so many birds to root for, why bother with grouse?

In derivatives

rooty (adj.)

1. (US) of a sporting contest, well- and fervently supported.

[US]World (N.Y.) 14 May 3/4: A closer, a more exciting or a more ‘rooty’ game of ball has never been played at Brotherhood Park.

2. see also under root v.