I haven't taught in a while, going on two years because I'm one of the privileged PhD students who is on funded research. I miss it from time to time because seeing a student understand a concept or learn they in fact can do something they were struggling with is beautiful, and this is what Assassination Classroom celebrates in its silly, heartfelt way
There is a darker side to it, beyond the turn it takes once the reality of the situation the students are placed in, that, yes, they do in fact need to kill their teacher (as they were told in the first episode/chapter). This is the more subtle implication of what the series means by "kill your teacher".
Teaching is demanding, especially teaching at the caliber depicted in the series with how much individual care and attention students get. In this day and age it's not sustainable. I say that from personal experience. I tried to be there for my students as much as I could when I taught my writing class for the first time, chasing students down when they were falling behind, giving as many opportunities as I could extend to help them catch up and practice their craft. It's rewarding but it is some of the most exhausting work you can do.
When Korosensei says he won't be teaching again the next year, this is what he's getting at. He is teaching at superhuman levels (literally) and even without the literal meaning of "kill your teacher" guiding the show, the implication is he can't keep doing that. Principal Asano also fits into this. Although he lasted a little longer than one year, he still has his approach to teaching change after the demands of society come for his first students.
Teaching is an incredibly valuable thing when done well and to do it well takes proper support beyond just being a good teacher. We as a society (particularly in the US) have destroyed what support was there and everyone is suffering for it