Since each vertex is labeled with a subset with two elements of $\{1,2,3,4,5\}$, then any permutation of $\{1,2,3,4,5\}$ is going to induce a permutation of the vertices. Moreover, if $\{a,b\}\cap\{c,d\}=\emptyset$, and $\sigma$ is a permutation of $\{1,2,3,4,5\}$, then $\{\sigma(a),\sigma(b)\}\cap\{\sigma(c),\sigma(d)\} = \emptyset$. That is: this permutation of the vertices sends adjacent vertices to adjacent vertices (and non-adjacent vertices to non-adjacent vertices).
Also, two permutations induce the same permutation of the vertices if and only if they are identical permutations (you should prove this). That means that every element of $S_5$ induces a distinct automorphism of the graph.
Are these all the automorphisms, or are there more? If $\tau$ is any automorphism, then by composing with an appropriate permutation of $\{1,2,3,4,5\}$ you may assume that the map fixes $\{1,2\}$; that means that the vertices adjacent to $\{1,2\}$, $\{3,4\}$, $\{3,5\}$, and $\{4,5\}$, must map to each other. Composing with an appropriate permutation (that fixes $1$ and $2$) you may assume that the permutation also fixes $\{3,4\}$; and composing again by a suitable permutation, you can make it fix also $\{3,5\}$ $\{4,5\}$ (again, you need to prove this). So then you are left with an automorphism that fixes $\{1,2\}$ and its three neighbors. If you can show that this is also given by a permutation of $\{1,2,3,4,5\}$, then you will have shown that every automorphism "comes" from an element of $S_5$ (since composing with suitable automorphism that do give you the identity).