In regard to iOS apps, this is not as relevant as it was before the app-to-app integration features of iOS 8.
Basically instead of just integrating another app/product with your own, you contact the vendor and ask them for a promotion on their blog, integrations page and e-mail list in return.
Hustle a little and let them know you have other integrations to pursue, if they’re reluctant to promote you.
It’s more about serving them, not you. Your product must add value to their customers. If your customers gets value too, that’s just great.
But your own customers is secondary here.
A good way to spot integration opportunities is to follow their blog and/or forums (provided they have any). People often submit feature requests, even in blog post comments and as forum posts. There's opportunity for lots of missing links in other products where you might be able to step in a solve a little problem that only helps the customer and also the vendor of the other app.
You’ll often see new feature announcements on a blog, in a separate category and the comments here often spin off of the announcement and ask for related features. If your product fills a gap, you have a great opportunity for an integration.
Rob Walling spoke briefly about it on a The Foolish Adventure Show podcast: http://foolishadventure.com/audio/how-to-market-your-software-with-rob-walling-fa128/ — about 40 minutes in.
Wade Foster of Zapier wrote a blog post about it here: https://zapier.com/blog/integrations-simple-marketing-developers/