Hi and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
It sounds like the internal storage of your iPhone is full. This internal storage is the storage of your iPhone came with it and it can't be upgraded, To free space, you need to delete unwanted pictures, videos, movies etc from your iPhone and also delete them from Recently Deleted album. So you will get free space.
You can also delete unwanted apps, messages with attachments to free space. You need atleast 1GB free for the proper working of the phone.
To see how much space is remaining, go to Settings, tap General, tap Storage and iCloud Usage. How much is available under STORAGE section ? If it is less than 1GB, then you need to delete more stuffs from your iPhone (1GB = 1024 MB)
You can download the photos and videos to your Mac and then delete from phone. Follow the steps below to download them to your Mac from iPhone.
Connect the iPhone to your Mac with the USB cable. Now go to Applications folder and open Image Capture application. Image Capture will recognize the pictures in your iPhone (select your iPhone in left column if needed). Select the pictures you would like to import, then select the destination (like Desktop folder, Documents folder) and click Import.
This will download the pictures to your Mac.
Another way is, connect the iPhone to your Mac and open Photos or iPhoto app. Then click on File menu and select Import > iPhone. Then import the photos. Once they are downloaded to your Mac successfully, delete them from your iPhone and delete from Recently Deleted album as well on your iPhone. Deleting videos taken on iPhone's camera will give you more space FYI.
If you have any difficulty on following the steps above, please let me know and I will walk you through it