It may seem strange to play a game by choosing actions randomly. But put
yourself in the position of Player A and think what would happen if you followed
a strategy other than just flipping the coin. Suppose you decided to play heads. If
Player B knows this, she would play tails and you would lose. Even if Player B
didn’t know your strategy, if the game were played repeatedly, she could eventually
discern your pattern of play and choose a strategy that countered it.
Of
course, you would then want to change your strategy—which is why this would
not be a Nash equilibrium. Only if you and your opponent both choose heads or
tails randomly with probability 1/2 would neither of you have any incentive to
change strategies. (You can check that the use of different probabilities, say 3/4
for heads and 1/4 for tails, does not generate a Nash equilibrium.)