Why was Mr. John Keating expelled from school?

First of all, it is not completely clear if there is anyone alone responsible for Neil's death and you might want to look into this related question to gain some further insights into this. But still, assuming Keating was not entirely responsible (which the answers there agree with), the question stands why he was fired.

You have to see the viewpoint of the school administration. If I remember correctly they, being rooted in their conservative ways, were to a large degree aversive to Keating's way of teaching self-determination and free will to those young men with his unconventional methods. So for them it was neither Neil's inability to confront his father, nor his father's strictness that drove him into suicide, but the "insubordination" and "fruitless ambitions" that Mr. Keating inspired him to.

And after such a big tragedy happened Keating was the best scapegoat to blame for Neil's death and firing him was maybe also to prevent further damage to the reputation of the school, as this suicide could become quite a big scandal. An additional factor may also be that Neil's parents (especially his father) of course also only saw Mr. Keating responsible (for exactly the above reasons) and maybe even applied some pressure to the school administration in order to get him fired.

So no matter if he was really responsible for Neil's death, he certainly was in the eyes of Neil's parents and the school administration and you are in fact supposed to be confused (or at least moved) by this injustice. The narrow-minded establishment has once again won against a single creative and idealistic man (yet his efforts were not completely fruitless, remember the "Oh captain, my captain"-scene).

(I should admit, though, that this answer is entirely based on my clouded memory of a movie I haven't seen for a very long time.)

When Keating passed on his book of poetry to be read at the meetings of the Dead Poets Society, I believe that that was sufficient grounds for his dismissal. He was condoning the creation of a secret and unsupervised club/organization that met off of school grounds. After Charlie Dalton received his paddling for revealing the existence of the D.P.S. in the school newspaper, Teacher Keating only mildly reprimanded him.

Now, the school may have very well just fired Keating for being a Maverick and as a scapegoat. Nevertheless, the administration have just cause to dismiss Mr. Keating.