I can't install Monterey on my MacBook, installation fails

Follow previous good advise - First

To add though.

The Age of this computer will Govern what newer version of macOS it will Qualify to install. 

Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence.

Does this computer appear as Listed and qualifying for an upGrade to a newer version of macOS ? 

If Yes, suggest having at least 50 GB of Empty Space on the drive.

It will allow the Full Version ( about 12.xx GB ) to 1 - download, 2 - expand and then 3 - commence to install.

If there is Third Party Security Software installed - remove it.

It will interfere with both the download and installation process.

The below links will help identify which version of macOS  this computer may Qualify to run.

macOS Sonoma is compatible with these computers

Computer Qualifying for macOS 13 Ventura 

Computer Qualify for the Monterey upgrade

macOS Big Sur 11 hardware requirements

Make Time Machine Backup  before doing any upGrades 

I've had better luck from the Safe Mode because I evidently had some 3rd party software that interfered with the downloading and installing so try the following:

#1 - boot into Safe Mode according to Start up your Mac in safe mode - Apple Support.

NOTE: Safe Mode boot can take up to 3-5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

#2 - go to the System/Software Update preference pane and begin the download.

#3 - run the installer from the Safe Mode.

#4 - or go to the App Store and download again the update installer and install from the Safe Mode.

Repair a storage device in Disk Utility on Mac

Disk Utility can fix certain disk problems—for example, multiple apps quit unexpectedly, a file is corrupted, an external device doesn’t work properly, or your computer won’t start up. Disk Utility can’t detect or repair all problems that a disk may have.

If you run First Aid on a disk, Disk Utility checks the partition maps on the disk and performs some additional checks, and then checks each volume. If you run First Aid on a volume, Disk Utility verifies all the contents of that volume only.

1. In the Disk Utility app  on your Mac, choose View > Show All Devices.
2. Note: If you’re checking your startup disk or startup volume, restart your computer in macOS Recovery, select Disk Utility in the macOS Utilities window, then click Continue. If you check your startup volume (Macintosh HD), make sure you also check your data volume (Macintosh HD - Data).
3. 
   
4. In the sidebar, select a disk or volume, then click the First Aid button .
5. If Disk Utility tells you the disk is about to fail, back up your data and replace the disk—you can’t repair it. Otherwise, continue to the next step.
6. Click Run, then click Continue.
7. If Disk Utility reports that the disk appears to be OK or has been repaired, you’re done. You can click Show Details to see more information about the repairs. Otherwise, you may need to do one of the following.

• • If Disk Utility reports “overlapped extent allocation” errors, two or more files occupy the same space on your disk, and at least one of them is likely to be corrupted. You need to check each file in the list of affected files. Most of the files in the list have aliases in a DamagedFiles folder at the top level of your disk.
    • If you can replace a file or re-create it, delete it.
    • If it contains information you need, open it and examine its data to make sure it hasn’t been corrupted.
  • If Disk Utility can’t repair your disk, or you receive a report that the First Aid process failed, try to repair the disk or partition again. If that doesn’t work, back up as much of your data as possible, reformat the disk, reinstall macOS, then restore your backed-up data.

If your Mac has a Fusion Drive and you see a flashing question mark or alert, see the troubleshooting section of the Apple Support article About Fusion Drive, a storage option for some Mac computers.

If you continue to have problems with your disk or it can’t be repaired, it may be physically damaged and need to be replaced. For information about servicing your Mac, see Find out how to service or repair your Mac.

In Terminal…

diskutil resetFusion

What has often worked for others is to use this Restore Method instead of CMD+r keys...

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

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Shift-Option-⌘-R

Install the macOS that came with your Mac, or the closest version still available.