Q:

Linear combination of DT unit impulse

Came across this example in class but I'm not sure how the expression $x(n)$ was derived.

A:

The plot that shows the sequence $x[n]$ is also called as the lollipop plot and it shows the individual samples of $x[n]$ with their weights and locations. 
Each lollipop represents a discrete-time unit-impulse signal, $\delta[n]$, which is located at the position it's shown and weighted by the value of the signal there. The superposition sum of all those impulses compose the signal $x[n]$.
Now, your signal has 3 lollipops located at $n = \{-1,0,1\}$ and weighted by $\{1,2,3\}$ respectively.
Since an impulse located at the position $n=d$, and weighted by $a$, is given by
$$ a \cdot \delta[n-d]$$
by using this, you can compose the signal $x[n]$ from its lollipops as:
$$ x[n] = 1 \cdot \delta[n-(-1)] + 2 \cdot \delta[n-(0)] + 3 \cdot \delta[n-(1)] $$  
$$ x[n] = 1 \cdot \delta[n+1] + 2 \cdot \delta[n] + 3 \cdot \delta[n-1] $$
Note that a generalization of this also yields the sifting-sum decomposition of any signal $x[n]$ as
$$x[n] = \sum_{k=-\infty}^{\infty} x[k] \delta[n-k] $$  

