An often confusing aspect of reading the payoff matrix in a game theory setting (at-least for new comers or those delving into the subject after a hiatus) is the confusion between the row players and column players. A more intuitive method maybe to keep track of the process of propensity of movement of player's states (as per moving in the direction of higher utility) using some sort of color coding to see the direction in which they move. The following figure tries to capture the same - and observe the point where the arrows meet is the location of Nash Equilibrium. I know it may seem too formal an approach to bring in a design aspect (colors and arrows), but then game theory itself is a formalism of something most people would say is common sense, intuition so a bit more formalism wont harm - let me know if this becomes more intuitive:

Generic background: As is evident a rational agent has clear preferences (i.e. states that he likes) and always chooses to perform the action with the optimal expected outcome for itself from among all feasible actions. A utility function (in the form of the above payoff matrix) is used to map out real world choices to quantitative numbers. These numbers can be seen to be levels of happiness of the agent in those corresponding states.

Intuitively, a Nash equilibrium is a stable strategy profile: no agent would want to change his strategy if he knew what strategies the other agents were following. Nash equilibria can be strict and weak, depending on whether or not every agent’s strategy constitutes a unique best response to the other agents’ strategies.