Your student ID identifies you as a student of CBS. You need to upload a photo of yours and apply to get your student ID printed for you. This will take a couple of days but afterwards you can use your student ID for the whole exchange.
The student ID also gives you access to the library after hours. However, you need to let it be activated for that purpose. The IT desk at SP can help you with that.
To get your student ID follow these steps:
Head to https://studentcard.cbs.dk/
Use your CBS credentials to log-in.
Upload a passport picture and follow the instructions of your website.
Your student ID will be ready after a week or so, and it can be picked-up at the IT desk at SP right in front of the library.