Log in to LinkedIn Developer Network with your LinkedIn account.

On that page click on the Add New Application link.

LinkedIn Developer Network

Fill the form and ensure all required fields (which has an asterisk. see the image below.

LinkedIn form for creating Applications

Enter your email and phone number in the Contact Info fields.

Tick the following checkbox in OAuth User Agreement: r_emailaddress, r_basicprofile, and r_contactinfo.

Note: the new LinkedIn application interface has no option for r_contactinfo. It's nothing to worry about.

LinkedIn OAuth Agreement

Enter your website url in the OAuth 2.0 and OAuth 1.0a Redirect URLs fields.

Note: in the new LinkedIn application interface, use the following as your OAuth 2.0 and OAuth 1.0a Redirect URLs http://yoursite.com/wp-content/plugins/profilepress/social-login/?hauth.done=LinkedIn

LinkedIn OAuth Agreement

Agree to the Terms of Service and click the Add Application button to create the app.

LinkedIn OAuth Agreement

After creating the application, a success page containing the app keys and tokens will be displayed.

LinkedIn OAuth Agreement

Copy the API Key and Secret Key to the LinkedIn field in Social Login settings page.