Do you get an error message when you tap on "Manage Storage"? If so, what does it say? If not, does the tap simply register (the Manage Storage button reacts) but nothing happens?
Looking at your iCloud Storage graph you have a lot of space being taken up by backups. My understanding is that those are not downloaded to your iPhone, so they shouldn't cause problems with your local storage, however it would be good to investigate what backups you have stored on iCloud. Unfortunately in order to do so you'd need to be able to access Manage Storage, and from there Backups.
Looking at your iPhone Storage graph the portion taken up by iCloud doesn't appear particularly alarming. Based on your original post I was expecting it to take, say, the space that's taken up by apps (red bar). It simply seems that you have a lot of apps on your iPhone, or apps that take up a lot of space. These might be games, or then video / streaming / podcast apps where you have a lot of audio / video content stored offline.
The iPhone Storage settings page will show which apps are taking the most space, and if you tap on them you'll see how much is taken up by the app itself ("App Size") and how much by documents stored in it ("Documents & Data"). If you spot an app with a lot of documents and data, you should open the app and see if you've simply forgotten to delete audio / video files (for example TV episodes) that you've already watched.
