Google have implemented a new permissions system in Android 6 and above, that means specific permissions can be denied or revoked from an app. If you've accidentally (or intentionally) denied Mendeley the Storage permission, it makes the Mendeley app effectively useless, as it can't sync any data from your library, download any PDFs to read, etc.
Follow the steps below if you want to enable these permissions and get Mendeley working again.
1. Open your device's Settings app.
2. Scroll down to Device section and tap Apps.
3. Find and tap on Mendeley in your list of apps (choose the All Apps filter if you don't see it in the list)
4. Tap Permissions.
5. Tap the toggle next to Storage to enable permission.
6. Relaunch the Mendeley app.