How did they get this table?
The point is that 


  The output of an LTI system to an input is a weighted sum of shifted versions of the impulse response.


You can see that the input is $$x=(1)\delta[n]+(-1)\delta[n-1]+(1)\delta[n-2]$$
The response to $A\delta[n-n_0],\, \forall n_0$, is $Ah[n-n_0]$. By the principle of superposition, the output to a sum of shifted $\delta$ terms is a sum of shifted $h$ terms. Here:

$$y[n]=(1)h[n]+(-1)h[n-1]+(1)h[n-2]$$
where $h[n]$ is $$ \cdots \quad 0 \quad 1 \quad -2 \quad 3 \quad -4 \quad 2 \quad0\quad\cdots$$