Presumably you found that the stabilizer H contains x = (1 3 2 4) and y = (1 2). Easily x4 = y2 = (1) and y-1xy = x-1. So H contains < x, y : x4 = y2 = (1) and y-1xy = x-1>. This last description is a presentation of a group via generators and relations. In order to understand this idea, you have to look at "free" groups. It turns out that any two groups with the same presentation are isomorphic. The dihedral group of order 8 is D8 = < r, s : r4 = s2 = (1) and s-1rs = s-1>. So D8 is isomorphic to a subgroup of H. Since this subgroup has index 3 in S4 and H is unequal to S4, H is equal to this subgroup.
So in sum, the way to prove generally two groups are isomorphic is to show that they have the same presentation.