Finding Nash Equilibrium
The two (pure) Nash equilibrium in this game is (Betray, Silent) and (Silent, Betray).
Let us see why (Betray, Silent) is an equilibrium.
Let us look at person A.
Person B is playing Silent.

If she plays Betray, she gets $2$.
If she deviates to Silent, she gets $1$. So she would not play Silent.

Now consider person B.
Person A is playing Betray.

If she continues playing Silent, she gets $0$.
If she deviates to Betray, she gets $-1$. So she would not play Betray.

No player has an incentive to switch strategies. So this is an equilibrium.
The same argument holds for the (Silent, Betray) equilibrium.