If you read the article to the end, it says that Switch-reference typically spreads by areal diffusion.
How does it originate? Well, I do not know about those languages, but I will try for fun, assuming there won't be a good answer anytime soon, because "typical" would require a lot of reconstruction, speculation, or for there to be only a handful of originating languages.
"If you eat I will drink" is a perfect example to speculate about, reason being passive voice, which is mentioned in the article, and it just so happens that German "werden" is used in both active future and passive present (which makes for a couple ambiguous corner cases by the way; e.g. ich werde vergessen "i am being forgotten" vs "I will forget", in which vergessen is exceptional, as it is homonym with its past-participle, which is no problem because the active form is regularly bi-transitive so that the second interpretation appears as a very unlikely ellipsis).
So, let's see "Wenn du ißt, werde ich trinken"; the passive would use a past participle, "ge-trunken", so there's no ambiguity; The funky bit is, if the first phrase is taken as the subject of the second, can a passive clause be constructed in another setting? "Wenn du nicht aufißt, wird's regnen" (if you not eat-up, then will't rain) is not exactly passive, it isn't medio-passive easier, but it gives linguists a headache and, importantly, it shows what I mean. It shows it especially well because the indefinite pronoun could as well imply that the implicit elided food will do the raining. Weather-verbs are mentioned in the article as being typically subjectless, but switch-reference marked nevertheless!
Overall, I guess switch reference arises just from the need to separate the second subject further away from the first verb in any word order. cp "(if he try s [it) could go wrong]". English morphology pretty much rules out misunderstandings because it inflects most pronouns, or, where it don't, inflects the verb. Yet we can optionally insert "then" for clarity. I'm a bit fixated on if because that appears out of nowhere in one of the article's examples, for a phrase that looks more like "you try I'll hit ya".