From following http://www.gametheory.net/dictionary/Games/SymmetricGame.html
How should the values in each row and column be interpreted ? A,B are states & (1,1) , (0,3) , (3,0) , (4,4) are actions ?
Is this a correct interpretation ? :
If player 1 is in state A and player 2 is in state B the payoff for player 1 is 3 and payoff for player 2 is 0 , therefore player 1 wins in this instance of game ?
It is better to think of the set $\{A, B\}$ as a set of strategies. When player $1$ and player $2$ play both strategy $A$ they each get a payoff of $1$. Or when player $1$ plays $B$ and player $2$ plays $A$, player $1$ gets $3$ while player $2$ gets $0$ and so on.