Children learn the meanings of hundreds of words before some of them (not all) start learning to read and use a dictionary. The idea of "definitions" in Webster's sense makes its impression and from then on we feel challenged to make definitions ourselves, to define our terms.
The formally educated discover a core flaw in many an argument is one or more inadequately defined terms, and in a debate situation, this weakness may sometimes be exploited in front of judges.
Wittgenstein's "language game" concept was on many levels designed to "push back" against the overwhelming impression a dictionary makes on our concept of "meaning". We get to mentally roll back to that earlier time in childhood development when we lived surrounded by meaning but without the benefit of a dictionary.
Apparently we learn to appreciate meaning well before we become cognizant that words have definitions. The "language game" concept is calculated to rekindle that understanding.
But is the concept of "language game" itself well-defined? Here is where you need to be a great writer like Wittgenstein to deliberately allow the notion of "fuzzy borders" into one's philosophy.
To an Anglo reader especially, "fuzzy" connotes "wooly" and "wooly-minded" means as if one's brain were but sheep hair, a neural net without the capability to sustain either rhyme or reason, at least in a given situation. When one's mind goes "all wooly", that especially is a time not to be doing philosophy.
Wittgenstein is a genius because he's regarded as sharp, focused, intense, right up through On Certainty, whereas in Philosophical Investigations he is already allowing concepts to remain willfully fuzzy, as when we say "the broom is in the corner" with just the right level of approximating inflection.
Is a trafficked intersection, a place where roads cross, a good example of a "language game"?
This question immediately takes us to the frontier in Wittgensteinian linguistic philosophy, as the bona fide linguists have an investment in not letting "language" become too distant from what tongues do, the dance of phonemes, the parade of words.
If stripes on asphalt, colored light signals, the motor vehicles themselves, are allowed to get sucked into "language", then their discipline (linguistics) would break down under the strain. Suddenly everything is language and it's like there's nothing more to say about it in general, that doesn't sound like philosophy, a different discipline.
So, for academic reasons, you will find many professionals backing away from the prospect of a street intersection being a "language game", even though computer simulations have us practice the many nuances, of signalling, of driving through red lights, of pedestrians, of billboards, even web cams. In England at least, a busy intersection is likely watched by web cams.
"Please oh please don't let them call that a 'language game'" many a logician will pray on his knees at night. Why? Because logicians do not want Logic to have to encompass transportation theory, intersection topology, knot theory, and whatever else may be involved (freeway design, concrete mixing, paving... any number of "dirty jobs").
All these professional considerations aside -- we get paid to think about X and so have an investment in X not morphing into some Y we feel more wooly-minded about -- Wittgenstein did philosophy a great service by putting the word "ball" and the bouncing sphere heading out from the car park, towards a busy street, on the same level.
Both the word and the thing have meaning.
A "word" tends to be some little ink painting or pixels etched in light, whereas the bouncing thing is more rubbery and free to move relative to other objects (words tend to be "frozen in place"). Perhaps because of these trivial differences, we forget that both items have a semantic role in whatever language game we're playing.
It's not as if a "word" were language whereas a physical ball were not -- unless of course we're linguists or logicians afraid of needing to play sports to earn a paycheck. A busy street intersection is very much a "language game" comprising a grammar.
The timing of the colored lights plays a big role in this grammar. You do not need a dictionary to figure that out. "Red means stop, green means go" -- that's just obvious. Our understanding is compelled to take that in, at great risk to our person if we can't show that understanding when it matters (when playing that language game ourselves, as pedestrians, as drivers).
As Wittgensteinians we're intellectually mature enough to accept a crossroads as a language game, and yet not affect the expert's tone about the subject matter of traffic engineering, even now that we're convinced that "crossing the street" is within philosophy's purview (as a "language event" i.e. a "move in a grammar").
We're smart enough (like rats in a maze are smart) to know that even if language is everything and everything is language, our philosophy need not be all things to all people.