UPDATED: July 28, 2016
Unfortunately, the sync feature in the LinkedIn iOS app is not currently functional. As a workaround, you can go to https://www.linkedin.com/people/export-settings and export LinkedIn contacts to a CSV or vCard file. That file can then be imported into any account (Gmail, Exchange, etc.) you currently have syncing with your iPhone. After those contacts have fully downloaded to your iPhone, launch CircleBack to bring them into your CircleBack contact list.
LinkedIn has restricted API use, limiting the information CircleBack and other apps can use. Therefore, the contacts must be first synced to your device. Once they are in your phone's contact list, CircleBack would be able to get those contacts from your device the next time you launch the app. These connections won't reflect as 'Networks' under your CircleBack app as the steps provided are the work-around to connect these sources.