Here, $$\begin{pmatrix}I/2 \\ X/2 \\ Y/2 \\ Z/2 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix}1 &0 &0 &1 \\ 0 &1 & 1 & 0 \\ 0 &i &-i &0 \\ 1 &0 &0 &-1 \end{pmatrix} \begin{pmatrix} |0\rangle\langle 0|/\sqrt{2} \\ |0\rangle\langle 1|/\sqrt{2} \\ |1\rangle\langle 0|/\sqrt{2}\\ |1\rangle\langle 1|/\sqrt{2} \end{pmatrix}.$$ The $4\times 4$ matrix (with the factor of $1/\sqrt{2}$ is clearly unitary. You can probably also find a unitary relating Paulis to $|i\rangle\langle j|$ for higher dimensions that are powers of 2, but I haven't thought about it.

See Theorem 8.2: (Unitary freedom in the operator-sum representation) in Nielsen & Chuang, which says that $\{E_i\}_i$ and $\{F_i\}_i$ are sets of Kraus operators for the same quantum channel iff $E_i = \sum_j u_{ij} F_j$ for some unitary matrix $u$.

it should have said $\{I/2,X/2,Y/2,Z/2\}$ (just edited the answer); does it make sense to you now?