External DVD Player Won't Connect

Make sure you have enable the Finder to Show CDs and DVDs on the desktop.

Finder > Settings > General > Show these items on the desktop.

The empty drive will not be displayed, but the CD or DVD, once inserted into the drive, should be displayed.

Is your version of macOS up to date?

You'll need to connect the external DVD drive to the Macbook through a powered USB hub. The drive is apparently requiring more power than the Macbook's built-in USB-C port can supply to it.

I have two different external USB optical drives that exhibit the same symptoms as yours on both an M1 MacBook Air as well as an M2 MacBook Air. Using a powered USB hub alleviates this issue. The powered USB hub requires a USB-C power connection (like from your MacBook's power adapter). You then plug your MacBook into one of the USB-C ports on the hub and the external optical drive into one of the USB-A ports as well. The hub will power both with enough power to run them properly. Simply using a USB to USB-C adapter does not work.

Do an Amazon search for "powered USB hub" for many results for around $30. I personally use this one.