I'm making this for two reasons, first, because I feel like Dan could do a better job of explaining why we do the things that we do on the regimen and I want to help those who are confused like I was, especially on red marks/hyperpigmentation, and second, because I want to check my own understanding of the regimen and make sure I'm not confused on anything myself.
I think the two main problems that people face are acne and red marks (which I especially had a hard time with). I think that acne is formed when bacteria and our skin oils meet, and so it is safer to kill the bacteria (bad guys) with benzoyl peroxide than try to fight our own skin's oils (good guys or neutral guys I guess). I think red marks come from skin trauma, either from acne or exfoliation or us not being gentle on our skin or using good moisturizer. I believe that red marks will go away faster if you are extremely gentle and use good moisturizer and protect yourself from irritation.
Cleanser: Basically, this is just to clear surface oils that could collect on our skin during the day, and make it easier for the benzoyl peroxide to get into the skin.
BP: Use 2.5% to penetrate deeper without causing as much irritation. (I believe the moisturizer cancels off the irritation that it does cause)
Moisturizer: To cancel out irritation from the BP and keep your skin in balance and protect it from trauma.
I think this is mostly correct.
"Red marks", are typically called "inflammatory" acne, meaning the skin is inflamed because of hormone imbalances, and will only become more aggravated with irritation. Not all acne can be cured topically, some people take pills that help balance out their hormones.
However, the regimen is a great topical treatment, because it's simple enough and it makes sense.
Cleanser - for washing your face, cleansing your skin from all the yucky build-up throughout the day.
BP - Benzoyl peroxide oxygenates the pores, killing the acne bacteria. It speeds up the skin's regenerative process, hence why people on the regimen get flaky.
Moisturizer - Since BP is really drying, you need to replenish it with moisturizer.