In some game theory books (e.g., Evolutionary Games and Population Dynamics), mixed strategies are described as points in a simplex. More specifically, for a game with $N$ strategies, from $R_1$ to $R_N$, mixed strategies will consist in playing the pure strategies with probabilities $p_1$ to $p_N$. Thus, a strategy corresponds to a point $\mathbf{p}=(p_1, \ldots p_N)\in R^N$ in a simplex.
My question is: Why is the notion of "simplex" invoked in some game theory books? Is there any important result in game theory that we can obtain by using the notion of simplex (that we couldn't obtain otherwise)?
The simplex is a well-understood object in mathematics -- it arises all over the place, from topology to optimization -- so saying "this thing we've described fits naturally in this context that you're familiar with" is a way to make mixed strategies seem more "natural".
One of the great theorems from topology is that a continuous map from the simplex to any simplicial manifold can be well-approximated (indeed, is homotopic via a "small" homotopy) to a map from a subdivision of the simplex to a simplicial decomposition of the target manifold; in short, something like the function that assigns the expected outcome to each mixed strategy can be well-approximated simplicially, meaning that you can make a pretty good representation of it on a computer by recording the values at finitely many "key mixed strategies" and then interpolating.
Whether that's of any interest to you, I don't know. I find it kinda interesting.