You're receiving a Permission denied error.
You believe this is because Laravel is trying to write something to the file system and doesn't have permissions to do so.
Change the owner or permissions of your storage path.
$ cd app/storage $ sudo chmod 777 * $ sudo chmod 666 */*
It's likely a user issue.
The instructions in the solution above make the storage directories world writable. The real issue is usually one thing.
Your console user and webserver user are different
This is normal, but often for development servers it's easier to make your webserver be the same user as your console user. The Fixing Permissions section of the two recipes listed below explain how to change this use for Apache and Nginx.