The English idiom “game on” is a phrase that speakers use when they want to express readiness for a challenge. This phrase is part of a wide category of sports metaphors that are popular in many English speaking communities. Many of these involve the word “game” as an abstract noun representing various sports, and in idiomatic use, various other activities.
The literal meaning of this phrase is a request to formally start or continue a paused game, and in figurative use, it is usually expressed alone. For example, someone who wants to participate in a simple race with another person might say, “you want to race me? Game on!” Since it is a stand-alone phrase, and not part of a larger sentence construction, the phrase is shown here as its own sentence when written.
In some other uses, English speakers might use the phrase differently, as part of an idiomatic verb construction. For example, someone who wants to start to be proactive about something might say, “I want to get my game on.” Here, the phrase refers to initiating something or becoming proactive or assertive. Some might even see the phase as a modern-day equivalent of a phrase used by the literary character, Sherlock Holmes, who was known for saying, "The game is afoot," at the beginning of a new adventure.
Besides this idiom, many other slang and colloquial phrases exist around the word “game.” English speakers might also talk about a “game plan” which refers to any kind of comprehensive plan, or say that someone does something “for the love of the game,” which means they enjoy that particular activity.
The phrase “game on” is part of an American interest in sports and recreation metaphors. Linguists have identified this type of idiom as a rather strong component of modern American speech. Newer uses, for example, include “hate the player, not the game,” which generally means holding an individual accountable for their actions rather than assessing those actions in the context of what is usually done in a given scenario.
The use of this sports metaphor is an abstract one. There’s also an opposite construction that gets used in modern English: “game over.” Where someone can use “game on” to express nearly any kind of beginning, “game over” functions similarly as a description of nearly any kind of end, and was likely drawn from 1980s era video arcade games, which commonly displayed this phrase at the end of a game. Both of these are extremely versatile phrases and are common in modern English use.