Purchasing the iCloud storage option.. this does not expand the storage capacity of your device. You cannot expand the storage capacity of your phone. iCloud is more of a 'syncing' service.
Here is how to check what is taking up all your space on your phone
How to check the storage on your iPhone, iPad, and iPod touch - Apple Support
If the majority of you space is being taken up by photos, and you are using iCloud Photos.. make sure you have the Optimize Storage option turned on.
Set up and use iCloud Photos - Apple Support
Otherwise if you are not using iCloud Photos, you can transfer the photos to your computer and remove them from your phone.
Transfer photos and videos from your iPhone, iPad, or iPod touch - Apple Support
You can check how much iCloud storage space you are using here:
Manage your iCloud storage - Apple Support
Note that when you delete photos/videos they go to a 'Recently Deleted' album where they may be classified as 'other'. To completely remove your photos from your device you would need to empty out this album.
You an also check the Files app and see if there is anything in the Recently Deleted area.