I applied to a handful of professor positions after grad school, but ultimately settled in government instead of academia. Most jobs, even at R1 schools, requested a teaching statement as part of your application package. Most graduate schools actually offered a 1 credit class in how to design and write your personal teaching statement for people who would be chasing teaching jobs.
My impression was that they just wanted to see that you had taken at least a cursory introduction to education, could use a limited vocabulary of relevant terms, and were aware that we don't teach by rote memorization and recitation anymore. They like to see people talk about their passion for teaching, their organizing principles, their social conscience, and their long term teaching goals. Sometimes people emphasize what they see as their particular teaching strength, say dealing with problem behaviors in the classroom or connecting with disadvantaged students or adapting dictated curricula to individual needs and circumstances.
In other words, it's mostly BS from the perspective of an engineer.