It means, get a belt and tie yourself to the desk.

buckle down is a slang meaning to seriously focus and apply oneself (such as to work or study). In other words, it is to concentrate, work hard, and commit to getting something done

For such a stern photo, Eliot is funny! He's right; the idiom means it's time to tighten your belt and forget about eating and fasten the doors and shutters and hatches shut tight because hard times have arrived, a storm, a threat. Good Q !

I'm not thinking of a belt for pants, but a seatbelt for cars, airplanes etc. So you fix yourself to your job (or whatever you need to do). There's another phrase with a related meaning: "strap oneself to the beast". Like a wild animal, you have to tie yourself to it and tame it... get control of the situation. "I had so many ideas last night, but I strapped myself to the beast and wrote it all down."