That is a very overworked word, because it has at least three distinct uses:
As a subordinating conjunction†
John said that he would play the role of facilitator.
As a relative pronoun†
Facilitator is the role that John played.
As a demonstrative adjective or pronoun†
a. That role is the one which John played.
b. That is what John did.
In your example, that is employed in use 3b: it acts as a demonstrative pronoun†, and its referent (the entity it refers to) is the noun phrase The role:
The instructor's role is that of a facilitator.
The instructor's role is [the role] of a facilitator.
While does not belong to this clause; it is a subordinating conjunction† which introduces the following clause and defines its relationship to the preceding main clause. Its literal meaning is "at the same time as", but it carries a contrastive sense as well, emphasizing that in this "learner-centered" approach the instructor's role is reduced and the students themselves perform much of his traditional teaching role.
† These are the terms employed in the traditional grammar which you are most likely to have encountered. If you have enjoyed a Thoroughly Modern approach to English grammar you may know these as 1. complementizer, 2. relativizer, 3. determiner, and 3b. fused-head determiner; and you may have been taught that while acts as a preposition.