BusyContacts pulls in the LinkedIn contacts from Apple Contacts. Check Apple Contacts to make sure your LinkedIn contacts appear there by clicking on the All LinkedIn group in the left sidebar. If they appear in Apple Contacts, they should appear in BusyContacts.

Important Note: macOS Mojave 10.14 has removed support for Social accounts. Thus, starting 10.14, BusyContacts only supports Twitter accounts.

If the LinkedIn contacts do not appear in Apple Contacts, you may be able to solve this by doing the following:

• Open System Preferences > Internet Accounts > LinkedIn, and click the minus (-) button to remove the account.
• Reboot your Mac.
• Open System Preferences > Internet Accounts > LinkedIn, and re-add your LinkedIn account and make sure Contacts is checked.
• Launch Apple Contacts to see if All LinkedIn appears in the left sidebar and if any contacts appear when clicking on it. There's a very good chance no contacts will appear initially and that it will take quite some time for the contacts to sync down from LinkedIn. If so, this would be a good time to take a break and leave Apple Contacts running and check again in an hour or so.
• Once the LinkedIn contacts appear in Apple Contacts, quit/relaunch Apple Contacts, then in BusyContacts select BusyContacts > Preferences > Accounts, click the plus (+) button and add your LinkedIn account.