To download an eBook to your Kindle, you will need to have access to a computer, as unfortunately there is not an Overdrive application for the Kindle. From your loans page on your computer, select "Read Now with Kindle."
You will get a pop-up that asks you to set your Kindle preferences. If you click "Set Kindle Preference" then you will only see the Kindle Book option for any borrowed titles. If you click "No Thanks," you will continue to be given all options for every title. This preference can be changed in your account's settings.
Once you set your preference, you will be taken to Amazon's log-in page, where you will be prompted to enter your username and password to Amazon. After doing that, you will land on the Amazon listing for the title, however, instead of an option to purchase the item, you should see an option that says "Get Library Book" in the right-hand corner of the page.
Once you select "Get Library Book" the title will be sent to your selected Kindle device the next time it syncs.