According to Songfacts, Tony Banks explained in an interview for Uncut magazine:

“The Knife“ was an important song for us to write, because quite a lot of what wed done up to the point was quite soft and acoustic. Peter (Gabriel) wrote what you might call the best bit. We started trying all this extra stuff not just guitar solos, but really structured instrumentals. A lot of the songs had quite simple starting points we allowed ourselves to stretch.

Mike Rutherford also said:

Rock theatre was a tag we ended up with. Lyrically, the things that Peter was trying to say in those days, no-one could hear what he was saying because the PA was so bad. So he was trying to act out the songs, so people would know what they were about and that was where the whole thing sort of started. We were painting images, moods, atmospheres so giving it a setting seemed to work well for us.

Peter Gabriel declared:

The lyrics for “The Knife“ were partly me being a public schoolboy rebelling against my background. I’d been heavily influenced by a book on Gandhi at school, and I think that was a part of the reason I became a vegetarian as well as coming to believe in non-violence, as a form of protest. And I wanted to try and show how all violent revolutions inevitably end up with a dictator in power.