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There are two types of storage on the iPhone. One is iCloud Storage and the other is device storage (that inbuilt device storage, that is the one came with iPhone like 16GB, 64GB,etc.). The device storage can't be upgraded. If it is getting full, then you have to delete unwanted movies, photos, videos, music, etc. from your iPhone.
You can upgrade the iCloud storage if your iPhone is asking you to get iCloud storage which you plenty of space.
Go to Settings > General and tap iPhone Storage. You will see your device storage info under the Storage section and iCloud storage under the iCloud section.
If you are running out of iCloud storage, then you can upgrade the storage from Settings > name at the top > iCloud > Storage & Backup > Buy Storage Plan
But here, it sounds like you are running out of Storage (i.e device storage), so you have to delete unwanted photos, videos, music, etc from your iPhone. That is the only way. Open Photos and delete the photos or videos from All Photos or Camera Roll. Deleting videos will give you more space than deleting photos. After deleting photos/videos, open the Recently Deleted album at the very bottom there and delete it from there as well. You can also optimize the photos stored on your iPhone if you are using iCloud Photos. Go to Settings > your name > iCloud > Photos and enable Optimize Storage there. You don't have to delete photos if you are using iCloud Photos.
Then open Music or Videos app and delete unwanted music, movies, etc.
Then go to Settings > General and tap iPhone Storage and this will give you an idea about what is taking up all the space. Do the recommendations showing there. Also, check the Messages section as well. If you send and receive iMessages with attachments, then the messages will take a lot of space. So open Messages app and delete all unwanted conversations/threads. Also, delete the attachments from the messages that you no longer need. See this article, click this link and follow the instructions ---> http://www.idownloadblog.com/2016/01/26/how-to-bulk-delete-attachments-from-the-ios-messages-app/
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