When a campaign is running you will see your backers' names and pledge amounts in your Backer Report, which you can get to through the creator tools menu to the left of your project page.
Clicking on any backer will show you more information. For successful projects, after you send a survey, their responses will appear in the Backer Report. It's also where you can leave a note to yourself and mark off their reward as sent.
Note: Credit card information is not shown.