Compute filter output with given values of $n$
Your impulse response is $h[n]=\delta [n] - \delta [n-1]$, so it basically takes a copy of the input and then substracts to it a delayed version of the former. When $x[n]=u[n]$, the output signal is:

$$y[n] = u[n] - u[n-1]$$

This means that we have a step function (i.e. it is equal to $1$ for $n \geq 0$ and $0$ otherwise), and we substract to it a shifted step (i.e. it is equal to $1$ for $n \geq 1$ and $0$ otherwise). Therefore, $\forall n \geq 1$ we are cancelling out the two signals ($1-1=0$). This leaves us with only one not-zero value, which happens to be at the origin. So the output turns out to be

$$y[n]=\left\{
 \begin{array}{ll}
  1  &amp; \mbox{if } n = 0 \\
  0 &amp; \mbox{otherwise }
 \end{array}
\right. \implies y[n]=\delta[n]$$