I am unable to find my DVD writable drive on my M1 MacBook Air

An optical drive itself will not show up in the Finder. Only discs will.

That is because an empty drive by itself has no filesystem. Only the discs that go in the drives contain filesystems. If you are trying to copy files from the Finder to a new, blank disc, you would put the disc into the drive, then copy files to the temporary burn area that appears, and finally, press the button to burn the disc.

Store information on CD and DVD discs on Mac - Apple Support

Today's Macbook's USB ports do not supply enough power (or current measured in amps/milliamps) to properly power on and maintain power for many external CD/DVD drives (one exception is the Apple SuperDrive). You will need a USB-C powered hub. The hub connects to an external USB-C power adapter. The external optical drive then connects to the powered hub and the hub connects to the MacBook's USB-C port. The power adapter powers the hub which in turn supplies power to both the external optical drive as well as the MacBook.

Here is an example of one that I use and is verified to work with two different external optical drives I own, one of which is an LG Slim Portable DVD Writer (model SP80NB60). These are drives that do not operate properly when connected directly to the MacBook's built-in USB-C ports.

Since this is a Mac notebook with an Apple Silicon chip, there's an extra security hurtle that you may need to clear to get the Mac to use the drive.

Allow USB and other accessories to connect to your Mac - Apple Support

If the drive was not "allowed" to transfer data, the Mac would provide power to it, but that's all.

Hold down the Option key and, from the Apple menu, choose System Information...

Click on USB.

Does the drive show up under USB in System Information?

If not, then the drive is not being recognized at all.

Assuming the drive DOES appear in System Information:

Are you placing a new and empty disc in the drive? What happens if you put in a disc that has content in it?

I will also add that when the DVD drive (LG Slim Portable DVD Writer (model SP80NB60)) I mentioned is plugged directly into the MacBook's built-in USB-C port(s) that it behaves as you describe. The power LED lights up briefly, the drive clicks faintly but nobody is home otherwise.

The use of a powered hub I mention resolves the issue completely.

Another way to ‘see’ the drive is to open System Settings and scroll down to the very bottom of the menu where attached devices are shown. If the drive is recognized there will be a CD entry along with your keyboard, mouse, and printers. If no then it is likely the drive you using is not compatible with the version of macOS you are running.