I think it's supposed to work like this: assuming more users == more revenue, then opening up and promoting the use of APIs creates alternative channels to acquire new users (partner apps have their own incentive to attract users etc). Where it can hurt is when the integrating apps either don't substantially move the needle on new user acquisition, or worse they cannibalize your user base by creating reasons for your users to NOT visit your site.

IMO though the API "shutdown" was shortsighted by Twitter. If people are using a competing interface instead of your own, then maybe you should fix your own rather than just shutting down the API. Of course that's easy to say as an outsider. I'm sure the truth is much more nuanced.