Converting DFA to Regular Expression Using State Removal
Think of it this way. You can reach 2 from 1 in two ways


$1\rightarrow 2$
$1\rightarrow 3 \rightarrow 2$


Then from 2, you can move to 1 by $2\rightarrow 1$. From there again, move to 2 from 1 in the above two ways. So this NFA can be expressed as an RE as follows:

$$RE = (1\rightarrow 2 + 1\rightarrow 3 \rightarrow 2) \cdot [(2\rightarrow 1)\cdot (1\rightarrow 2 + 1\rightarrow 3 \rightarrow 2)] ^*$$

Now its just the matter of finding the REs for $1\rightarrow 2$, $1\rightarrow 3 \rightarrow 2$ and $2 \rightarrow 1$.