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.