The most probable reason why the campaign isn't sending any emails to the new contacts yet is that its status is already completed before you added the contacts. If a campaign has been completed, it won't do anything anymore. 

If that's the case, what you'll need to do is to resubmit the campaign again. So just edit the campaign, go through all the steps without changing anything, and then click Submit at the last step. Your campaign will be rerun and will send emails to the new contacts that the campaign recognizes. That means that the campaign won't send the emails to the contacts that the campaign has already sent emails to beforehand.