"The Dead Poet's Society" was a movie about John Keating. Keating was an inspirational teacher and this was Robin Williams portrayal of this amazing teacher. He actually taught in Nashville, TN for a short time and I got to hear him speak to the topic of education and today's youth. Keating is a motivator and very engaging. He makes you want to go into your classroom and challenge your students to challenge themselves.
"The story is set at the fictional Welton Academy in Vermont The script was written based on the author's life, Samuel F. Pickering, at Montgomery Bell Academy, an all-boys preparatory school in Nashiville,Tennessee"
"He was their inspiration. He made their lives extraordinary." simply means that no student ever forgot that they had sat in this man's classroom. These young men and women are at a very impressionable age and Keating has a way of touching their soul and earning their loyalty. He "gets" them. He has a unique and non-conforming way of reaching nearly every life he touches in the classroom. His unique style certainly makes administrator's uncomfortable. Can you imagine if you had walked into a stuffy English class and your teacher told you to rip out the first two chapters of the book and throw it away? I bet he/she would have immediately have become an inspiration. I hope some of my students feel that way about me.
Well, it refers to Robin Williams' character, John Keating. Think back to the scene where he asks the boys to climb up onto the centre desk in the classroom, in order to see the world from a different perspective. One of the things he shouts to them is "make your lives extraordinary" - and there is no doubt that Mr. Keating does inspire the boys to make their lives extraordinary.
Keating inspires Neal (played by Robert Sean Leonard) to act in "A Midsummer Night's Dream", and find the creative output which really sets him on fire and gets him excited. He inspires the boys to read poetry, and re-start the "Dead Poets' Society" which he himself was a member of as a boy.
So my reading of the tagline would be quite straightforward: Keating inspired the boys to be creative, and told them - pushed them - to make their lives extraordinary, creative and different.
Hope it helps!