Well, the first thing is that Kindles -- the eInk devices -- don't use ePub. You have to download the kindle version, if available.
If the book is only available as an ePub, you need an app on your phone or tablet that will let you access the file. If you're using a Fire, you still need an app; the built in eReader function of a Fire is Kindle based and doesn't do ePub.
Overdrive has an app -- and it's available via Amazon's app store -- but there are others as well. I read exclusively on my Kindle so if there's no kindle version available I don't borrow the book, but it seems to me the Overdrive app would be the easiest option as the borrowing should be able to be done directly through the app. So you don't have to worry with figuring out where you downloaded it to and how the app will find the file.
