There isn't much to analyze. Your root filesystem was configured to allow up to 6,553,600 file to be created on that filesystem and you have 6,553,600 on that filesystem.
You free an i-node by removing all of the hard links to a file.
You haven't told us what filesystem type you are using for your root filesystem. Some filesystem types automatically allocate i-nodes when needed; apparently the filesystem type you're using is not one of those filesystems.
You haven't told us which Linux distribution you're using. Some distributions have administrative utilities to reconfigure the number of i-nodes allocated in some filesystem types. If your distribution doesn't have a utility to do that for the filesystem type you're using and you can't delete a large number of files from your root filesystem; you'll need to make a backup copy of your root filesystem, boot from an alternative device, recreate your root filesystem with more i-nodes configured, restore the files from your backup, and reboot onto your new root filesystem.