Similar questions have been asked on this site, including by myself, but none of these have been given a satisfying answer.
The question is: Why does the Grassmannian scheme represent the Grassmannian functor? I have seen many books and articles about this, and they all treat it as an exercise to the reader. I am willing to admit that I may be too stupid for the exercise, but is there a textbook or survey article that explains this in détail? I mean it is somehow obvious, but I'd like to see it worked out. Please, if possible, no stacks-project or EGA.