I will give two explanations for why games crop up so fruitfully, one explanation from mathematics and the other arising from evolutionary biology.
The mathematical explanation for why games crop up in mathematics is that the truth of a complex statement
$$\forall x_0 \exists y_0 \forall x_1 \exists y_1 \cdots \varphi(x_0,y_0,x_1,y_1,\ldots)$$
is easily thought of as a game, where the first player (called Adam for $\forall$) plays $x_0$, challenging Eve (for $\exists$) to respond with a value $y_0$, for which Adam plays $x_1$, to which Eve responds with $y_1$ and so on, with Eve winning when $\varphi(x_0,y_0,x_1,y_1,\ldots)$. The point is that
(in the finite case) the statement above is true exactly when Eve has a winning strategy allowing her to defeat all plays of Adam. Thus, the truth of the statement with alternating quantifiers can be thought of in terms of these games and whether they have strategies.
Such a perspective is fruitful even when there are only a few quantifiers, as our calculus students learn when studying continuity or when seeing $\epsilon,\delta$ proofs for the first time ("you pick $\epsilon$, and I can respond with $\delta$").
Thus, the answer to your question about why games are so fruitful is that the existence of strategies for those games is intimately connected with the truth of complex statements. Since it is these statements that we are really interested in, it is fruitful to investigate them particularly with the game perspective.
One can prove using these ideas that all finite length games are determined, in the sense that they have a winning strategy for one of the players. That is, either Eve has a strategy that allows her to defeat all plays by Adam, or else Adam has a winning strategy allowing him to successfully challenge any play of Eve.
The natural generalization of these ideas lead to the Axiom of Determinacy, which asserts even that infinitely long games are determined. In this context, one imagines an infinite game, with Adam playing $x_n$ and Even playing $y_n$ for all natural numbers $n$, and then Eve wins if the play $(x_0,y_0,x_1,y_1,\ldots)$ is in the payoff set $A$, a set of infinite sequences. The Axiom of Determancy asserts that for every set A, either Adam or Eve has a winning strategy, and this axiom can be thought of as an infinitary de Morgan law:
$$\neg[\forall x_0 \exists y_0 \forall x_1 \exists y_1 \cdots A(x_0,y_0,x_1,y_1,\ldots)] \iff\exists x_0 \forall y_0 \exists x_1 \forall y_1 \cdots \neg A(x_0,y_0,x_1,y_1,\ldots)$$
which expresses the idea that if Eve does not have a winning strategy, then Adam does have a winning strategy. Synatically, this equivalence looks quite natural, since we are used to pushing $\neg$ through all the quantifiers and changing them to the dual quantifier. But in the infinitary context, this equivalence is not a matter of logic, and actually contradicts the axiom of choice. But it does suggest AD as a natural axiom. From AD one can prove that every set of reals is Lebesgue measurable and many other natural regularity properties, such as the property of Baire and the perfect set property. The consistency of AD over ZF is equivalent to the consistency of infinitely many Woodin cardinals, a large cardinal property.
Evolutionary biology. But let me come to another more speculative explanation of why games crop up so frequently and fruitfully. This reason has to do with evolutionary biology. The fact is that the truth of a statement involving many alternations of quantifiers is amazingly complex mathematically. Even definitions involving five or six quantifiers are rather complex, and give rise to very subtle distinctions in mathematics, such as the difference between uniform continuity and continuity, or families of continuous functions versus equicontinuous families of functions. Nevertheless, because of the way that we humans evolved, we are used to thinking strategically, in making decisions that are provisional based on the future actions of other individuals. Thus, the strategic way of thinking allows us more easily to understand the meaning of these complex statements. I would say that this facility with strategic thinking means in a sense that we may have a hard-wired kind of inherent ability to reason about extremely complex mathematic statements, since these statements can be equivalently expressed as strategic games.