If $I$ and $J$ are the strategy sets for 2 players respectively.
Let A = $[a_{ij}]$ and B = $[b_{ij}]$ be the random payoff matrices of player 1 and player 2 respectively.
Let X be defined as \begin{equation} X=\left\{x \in \mathbb{R}^{m} \mid \sum_{i \in I} x_{i}=1, x_{i} \geq 0, \forall i \in I\right\} \end{equation} representing the sets of mixed strategies of player 1, and Y be defined as: \begin{equation} y=\left\{y \in \mathbb{R}^{n} \mid \sum_{j \in J} y_{j}=1, y_{j} \geq 0, \forall j \in J\right\} \end{equation}representing the sets of mixed strategies of player 2.
Then, for each (x, y) ∈ X × Y the payoff of player 1 given by $x^TAy$.
I cannot seem to wrap my head around the last sentence. Is $x^TAy$ some sort of expected value for the payoffs? Please help, thanks.
Yes it is expected payoff.
Pick a term in $x^T A y$ like $x_i a_{ij} y_j$ this is the payoff for player 1 when 1 plays $i$ and 2 plays $j$ which is $a_{ij}$ then weighted by the probability of that happening which is the product of the two independent probabilities $x_i$ for 1 to play $i$ and $y_j$ for 2 to play $j$.
Add these all up and you get an expected value for the payoff to 1 when 1 is using mixed strategy $x$ and 2 is using mixed strategy $y$.