“Game on!” means that a challenge has been accepted and a competition can begin. “Game on!” supposedly was influenced by video games in the 1980s, when a game was often paused, and it was either “game on” or “game over.” However, citations from the 1980s and 1990s about the video game use appear to be lacking.  The term “game on” is said to have originated in darts.
 
“Game on!” became widely known by the 1990s.

 
(Oxford English Dictionary)
colloq. (orig. Darts). game on (also it’s game on):  (a) used to signal that play can begin, or that competition has resumed (esp. in earnest), in a game, match, or (in extended use) some other contest;  (b) used to indicate that one is equal to a challenge or ready for an activity: ‘bring it on!’
Recorded earliest in extended use.
1973 Sat. Night Apr. 40/3   Unprintable words were flung. Older brother clouted her… Banished to rooms upstairs… And then the first inevitable trespass. And it’s game on again.
1980 Observer 10 Feb. 44/3   It’s game on and Lazarenko is on the ‘oche’—the mark from which to throw.
1991 M. Myers et al. Wayne’s World (film script) 58   A car approaches. Garth moves the [hockey] net out of the way. The car passes. Wayne (shouting) Game on! Garth moves the net back. They resume playing.
1999 FHM June (Best of Bar Room Jokes & True Stories Suppl.) 39/2   She soon invited me back to her place for the other. Game on!
 
Wikiquote: Wayne’s World
Wayne’s World is a 1992 comedy about two slacker buddies and their local-access cable TV show.
(...)
Dialogue
While playing street hockey…
Wayne Campbell: Car
Garth Algar: Car!
Car drives past.
Wayne Campbell: Game On!
Garth Algar: Yeah, Game On!
 
14 January 1995, Palm Beach (FL) Post, ‘Game on!’ Players officially OK deal” by Brian Biggane, pg. 3C:
It was almost four hours later that Commissioner Gary Bettman and NHLPA Executive Director Bob Goodenow made the official announcement, donning hats bearing the phrase “Game On!”
 
12 May 1998, St. Louis (MO) Post-Dispatch, “Pronger’s injury is heart-wrenching for players, everyone else” by Tom Wheatley, pg. C1:
“When they say Game On, it’s game on again.”
 
26 April 2000, Philadelphia (PA) Inquirer, “Schilling winds up rehab,” pg. E1:
“Now it’s game on.”
 
Google Groups:: alt.revisionism
HEY NEO… LADS… IT’S GAME ON AT LAST.
susie
7/23/00
 
OCLC WorldCat record
Game on!
Author: Pat Doyle; Michelle Harkness
Publisher: Champaign, IL : Human Kinetics, ©2001.
Edition/Format:   Book : English
Summary:
77 games and activities for kids 5-14.
 
Urban Dictionary
GAME ON
A phrase said before preparing to engage in a competitive event.
Ryu sat down after winning the fight. When Ken’s name was announced for the next match, he stood up, winked at Ryu, and said with a grin, “GAME ON!”
by Joshiro007 Feb 12, 2003
 
Google Books
Cassell’s Dictionary of Slang
By Jonathan Green
London: Cassell
2005
Pp. 564-565:
game on! excl. [1980s+]
1 an act of excitement or anticipation, usu. about a possibly sexual conquest o when a night out or drinking session is arranged.
2 an excl. of triumph at having sorted out initial arrangements, [darts jargon game on! the game is about to start]
   
English Language and Usage
What is the origin of the phrase “game on”? Can it be used in formal conversations?
(...)
This is a common phrase, but I don’t think it is appropriate for formal conversations. Several dictionaries list it as slang or informal. I read this article that speculates that it might come from video gaming” “The literal meaning of ‘game on’ is a request to formally start or continue a paused game.” It’s counterpart being “game over.” – JLG Jun 5 ‘12 at 14:21