G does act on the set X by conjugation, since that is how it was defined. So does any subgroup of G, including H.
You don't find what part about your question self explanatory?
1. Let X be the set of G-conjugates of H, let [e],[g_1],..,[g_r] be a complete set of representatives of the conjugates (i.e. [g] is the conjugate gHg^-1, and inparticular [e] stands for the conjuate H of H).
2. H acts on X.
3. H sends the element [e] in X to [e], so there is at least one fixed point.
4. By the class equation for the action, this implies that there are at least p-1 fixed points.