The key word is 'given'.
Given that the Entrant stays out, the game terminates with payoff (Entrant,Incumbent) = (0,2). Given that the Incumbent fights, if the Entrant enters the payoff is (-1,-1) compared with the adopted strategy 'Stay out' giving a payoff of (0,2). So the Entrant's payoff falls from 0 to -1 by changing the strategy 'Stay out'. The given strategies do give a Nash equilibrium.
Look at the strategies Entrant stays out, Incumbent acquiesces if Entrant enters. Given that the Incumbent acquiesces, if the Entrant stays out their payoff is 0 but if they enter their payoff rises to 1. This is not a Nash equilibrium as Entrant can increase the payoff by changing their strategy.
Eric Rasmusen has a test on Nash equilibrium on http://php.indiana.edu/~erasmuse/GI/index.html under 'Miscellaneous'.It is trickier than it looks!