The word "unique" is misleading here. What is meant is that there is a one to one correspondence between partitions of a set and equivalence relations on it. Each partition induces an equivalence relation, and each equivalence relation partitions a set. It looks like he leaves part of this as an exercise for the reader.
Sketch of Proof:
An equivalence relation partitions a set:
The equivalence classes form the disjoint "pieces" of the partition. Note that
a.) every element is in some equivalence class (namingly its own).
b.) The equivalence classes are disjoint.
A partition induces an equivalence relation:
Define two elements to be "equivalent" if they reside in the same "piece" of the partition. Show that this relation is reflexive, transitive, and symmetric.