If your Switchboard is loading other widgets, but not the contents of the Call Activity Widget, verify that the Internal Directory is setup:
Go to Tools > Internal Directory and check that there is a group with contacts in it (such as Default Group). Is there a Selected group that contains contacts? If there is a group with contacts in it, try deleting the group from the Internal Directory > Selected box and and then re-adding it.
If issue persists, reassign the Switchboard Layout to the extension having the issue and verify it is active:
Go to Setup > Manage > click on yellow pencil > Assignment tab > Switchboard Layouts tab > verify the Layout is assigned to the extension and that it is Active.
Next, check the extended Permissions under Setup > Permissions to verify the extension has permission for: Can View Calls.
Finally, try a different browser, or private (or Incognito) browsing window. (under File menu of your browser). If your browser is causing the issue, you should clear your browsers cached files and cookies.