How do I delete old submissions, in order to free up space?

You can only free up the Upload space. These are submissions, which have file attachments. So, you can delete some of these submissions and free up the Upload space. But, please have in mind that deleting submissions is irreversible and you will not be able to restore them later.

In order to better understand your account usage and limits, you can check this guide:

https://eu.jotform.com/help/408-Understanding-Your-Account-Usage-and-Limits