How do child themes work?

Child themes are different then parent themes (in this case Twenty Fifteen is the parent theme). Child themes are just copies of your parent themes files. Then you modify those child theme files and they override your parent theme.

The advantage of doing this over creating a new theme is that you can update the Twenty Fifteen theme without affecting anything. If you created a new theme you would need to modify it manually to add the Twenty Fifteen update. Also, if you just modified your parent theme when you update you would lose off your customizations.

Will I have total control over the theme by just creating and applying a child theme? Or Will I be restricted in some way?

Child themes do not restricted anything. If anything it actually frees you up.

Have you read Child Themes?

What is a great question. Have you read the Codex page on Child Themes? If not it will answer any other questions you may have.