Well, basically, you can just add to the story after reading their Writer Guidelines (these are shown on the "Add Chapter" page)
This only works for moderated and public stories. It isn't possible on private stories.
If the story is public, your chapter will be shown immediately.
If the story is moderated, the story owner (or an editor) has to approve the chapter. Most story owners appreciate additions and will approve them. If the quality of their writing doesn't meet their standards, they might edit your work or decline it and ask you to revise it.
So it could also be an advantage to ask a story owner first.
And I think it is just as simple like
Hey [name of the author],
I really enjoyed your story [name of the story] and would like to add a [chapter/branch] to [it/name of a chapter].
[Do I have to pay attention to something in particular? / Do you want me to recommend a chapter to add on? / Do you have a branch which you would like me to add on? / What direction would you like me to explore?]
I think there are more users on the site who would like others to add to their stories than users who actually add to other authors' stories, so you shouldn't have any problems with that
Depending on what you exactly want, you could also ask them for sth like if they would edit your chapter, give you advice, collaborate with you on a chapter and so on.
(If you choose a story where the owner wasn't here for a long time, you can ask on the forum for approval by a moderator or adopt a story like you already did.)
