My iPad system storage is using up most of my space. How can I check what is being stored?

You can't see the system storage content, since it is automatically managed by the iOS/iPadOS and contains system files that are necessary for operating the core system of the device (mostly latest updates).

The same thing happens for "Other" category, mostly filled with cache and system files created by Safari, Mail app, other apps and iOS/iPadOS itself.

You can't interact with those "storages".

That said, update your device to iOS/iPadOS 13.6.1 or later, as per this support article: Clear Other storage on your iPhone or iPad - Apple Support

If you already updated your device to 13.6.1 or later to no avail, some benefits to reduce that space can be also obtained creating a full backup of your device, then restoring it to factory settings and, in the end, setting it up as new, as per these instructions: Restore your iPhone, iPad, or iPod to factory settings – Apple Support

When done, restore data from your previously made backup, then restart your device and check your device storage again.

If the issue persists, contact Apple Support directly, using this link: Get Support

Regards

Giulio

If your internal storage is full, your only option is to better manage the storage that you have - and delete unused Apps or unneeded data. 

These support pages should prove to be helpful:

What's the difference between device storage and iCloud storage? – Apple Support

Manage your iCloud storage – Apple Support

How to check the storage on your iPhone, iPad and iPod touch – Apple Support

Clear Other storage on your iPhone or iPad - Apple Support