To tweet from Nelio Content you have to authorize Nelio Content to access your Twitter account. Go to the plugin’s Settings and click on the Twitter icon:
When doing so, a new window pops up and asks you to authorize Nelio Content to use your Twitter profile.
If you didn’t have your Twitter session open, Twitter will ask you to log in first in order to grant Nelio Content access:
In both cases, after clicking on Authorize App, your list of Connected Profiles will end up including your Twitter profile: