Yep so when your Sandbox account is provisioned it is initially in a status of Offline. The first time you use it, you will need to click that button "Copy Data to Sandbox"
This moves a copy of your production account to your sandbox account. Once the data is moved over it will go green and you should be able to log in.
When you click that copy button there is an options for "access after sandbox refresh"
If "all users" is selected then every user in your production account will have access to the sandbox. If you leave it as Administrators only, then only users with the Admin role will have access. At which point you can assigned anyone without admin role access as needed.