Entry in progress—B.P.
 
fan
       
Wikipedia: Royal Rooters
The original Royal Rooters were a fan club for the Boston Red Sox in the early 20th century. They were led by Michael T. McGreevy, who owned a Boston saloon called “3rd Base”. While M.T. “Nuf Ced” McGreevy was certainly the spiritual (in both libations and foundations) leader of the Royal Rooters, Boston Mayor John F. Fitzgerald, the maternal grandfather of John F. Kennedy, served as chairman for a while, and during that time, M.J. Regan was the secretary. Other members included C.J. Lavis, L. Watson, T. S. Dooley, J. Kennan, and W. Cahill, among others. Their theme song was “Tessie” from the Broadway musical “The Silver Slipper”. Though the musical ran for less than six months, the song has gone down in history. The Rooters sang “Tessie” at games to encourage their Sox, while simultaneously distracting and frustrating the other team. They were especially important in the first World Series, in 1903, when the Red Sox played the Pittsburgh Pirates. The Royal Rooters would go to Pittsburgh and sing Tessie to distract the opposing players, especially Honus Wagner. Therefore, after falling into a 1-3 deficit, Boston rallied to win the Series with four straight victories. The original Rooters disbanded in 1918.
 
Their spirit lives on via the current version of the Royal Rooters represented within a group known as Royal Rooters of Red Sox Nation. The current Rooters are based in the Boston area and meet informally for Red Sox games as well as for “outings” in various locations around the country. There is a fairly large contingent in New York City, and their base has been the Riviera Café (known as “The Riv”) in the West Village.
 
Wikipedia: “Take Me Out to the Ball Game”
“Take Me Out to the Ball Game” is an early-20th century Tin Pan Alley song which became the unofficial anthem of baseball although neither of its authors had attended a game prior to writing the song. The song is traditionally sung during the seventh-inning stretch of a baseball game. Fans are encouraged to sing along.
(...)
[Chorus]
Take me out to the ball game,
Take me out with the crowd;
Buy me some peanuts and Cracker Jack,
I don’t care if I never get back.
Let me root, root, root for the home team,
If they don’t win, it’s a shame.
For it’s one, two, three strikes, you’re out,
At the old ball game.
 
Merriam-Webster Online Dictionary
Main Entry: root
Pronunciation: \ˈrüt also ˈru̇t\
Function: intransitive verb
Etymology: perhaps alteration of rout
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
 
(Oxford English Dictionary)
root, v.
colloq. (orig. U.S. slang) To cheer for a (baseball, etc.) team. Also transf., to be active for a person or thing by giving support, encouragement, or applause. Also without const.
1889 N.Y. Semi-Weekly Tribune 5 Nov. 5/4 Murphy has done little but ‘root’ for the Giants this year.
1895 in Funk’s Standard Dict.
1895 J. S. WOOD Yale Yarns 152 We rooted hard, too, and did a lot of shouting and yelling.
1897 FLANDRAU Harvard Episodes 164 The fellows who had promised to vote for Wolcott..were beginning now to ‘root’ for him vigorously.
1922 S. LEWIS Babbitt v. 66 Zilla keeps rooting for a nice expensive vacation.
   
(Oxford English Dictionary)
rooter
colloq. (chiefly U.S.) One who cheers or ‘roots’ for a (baseball, etc.) team. Also transf., one who supports or encourages another; a warm advocate, a partisan.
1890 N.Y. Press 8 July 6/1 At this juncture the New York rooters began to ‘pull’ for the home team, but the effort was useless, not a man..succeeded in reaching first base.
1895 in Funk’s Standard Dict.
1901 Speaker 19 Jan. 439 At the first class cricket matches for years he has been what in the States they call a rooter.
1901 Daily Colonist (Victoria, B.C.) 31 Oct. 4/3 ‘What makes him look so very white?’ inquired the fairy maid. ‘He’s had the starch knocked out of him,’ the woolly rooter said.
1931 L. STEFFENS Autobiogr. II. III. xxxiii. 593 They..don’t ask about, they don’t hear of, the always existing few quiet students with concealed gifts in the rooters at a football game.
 
9 September 1889, Kansas City (MO) Times, pg. 2:
A NEW THING IN BASE BALL.
“Rooting” as a Factor in the Brooklyn Club’s Success.
[Brooklyn Eagle.]
About a month ago the Eagle printed a letter from “One of the Players” of the Brooklyn base ball team. At that time the team was behind St. Louis very considerably and showed no signs of doing better then taking second place. “One of the Players” suggested that the real reason of the home club’s failure was the entire absence of “rooting” among the Brooklyn spectators. He called on all good Brooklynites to “root” for the home team. At that time Brooklyn was behind and doing badly. Evber since then she has played such a stiff game that the redoubtable St. Louis browns could not retain their lead. “One of the Players” in the following letter claims that to him is largely due the credit for the Brooklyn’s present position in the race:
 
To the Editor of the Brooklyn Eagle.
I think I can depend on your justice and sense of right to bear me out in the claim that to my suggestion is due the present proud position of the Brooklyn base ball club. In a letter to the Eagle, written a month ago, I made the statement that the reason Brooklyn could not get the lead was not that there was anything wrong with the team, but that there was something very wrong with the spectators, a large number of whom were New York people. I showed that, while the people of all other association cities “rooted” for the home team from beginning to end of the game. Brooklynites did not do this as a rule, many being indifferent and some actually hostile to the success of the home club, thus dividing the “rooting” and forcing of the Brooklyn boys to depend on play alone to win their games. I pointed out the injustice of this and counseled all Brooklynites to “root” for their own.
 
The effect of my advice was immediately observable. Brooklyn struck a winning gait at once and gained twenty points on St. Louis in a single week. Then she went west and lost ground again, but on the whole she did better than on any other western tour this season. She dropped three straight games in St. Louis and inexperienced people here thought it was all up with the Brooklyns and that the browns would be champions again. I knew better, though, and so did those who understand the power and effect of good judicious rooting. The reason that Brooklyn made such a slum, was that she was away from home and out of range of those who were rooting for her, as they were new at the business and could not root as far as St. Louis. Besides, many of them forget that they had to root when the team was away. The proof of this is found in the sudden jump which the team made for the lead when she got back on her own ground.
 
If the rooters of Brooklyn stick to their work they will certainly win the pennant for our city. This is all that I esire. For myself I ask nothing. Yours modestly,
ONE OF THE PLAYERS.
 
Forty years ago “One of the Players” would have been laughed at for claiming that the spectators of a ball game could aid or restrain the players by any concerted mental effort, but time goes on and science progresses. The recent hypnotic congress in Paris listened to more wonderful things than that from the mouths of the foremost scientists of France, Belgium, Holland and Switzerland. The power of mind over matter is known to be wonderful and far reaching. It is not yet fully explored and no one knows what its limitations may be. Therefore there can be no harm in the spectators of ball games in thsi city “rooting” on, provided always that they “root” with prudence and judgment and not indiscriminately. In time they will probably be able, by strict attention to business, to project psychic force as far as Kansas City in such quantities as to insure victory for Brooklyn even at that distant point.
 
Google Books
April 1897, Kansas University Quarterly, vol. 6, no. 2, “Dialect Word-List - No. 4” by W. H. Carruth and Paul Wilkinson, pg. 91:
rooter: an enthusiast at a game.—General.