There are several explanations of pull-up resistors that I find gloss too much over the details. After some researching and doing circuit analysis to clear up how they work (theoretically), I wrote a blog post. Maybe it's useful to others as well.
This post uses quite some formulas, so I'll provide a link instead:
https://blog.mh-nexus.de/2018/04/how-pull-up-resistors-really-work/
I edited it over many days, adding images and changing the wording. If there is a mistake now, please let me know.