While setting up a new campaign, you need to add people to the campaign either by clicking the Add People button or via the red "+" button if there is at least 1 contact in the campaign already:
Once you click Add People, you see a pop-up with 3 options:
Select one of these and add people to your campaign.
When you add people from existing, the prospects listed are all available to be added to the campaign, which means that any contacts associated with a different campaign will not appear. Select the desired people and add them to the campaign.
If you need to add new contacts to an existing campaign, simply go to your Campaign - People tab and use the red button to add more people: