Easiest ways that would not involve drilling into the slide out to fix this.
Just glue and attach with screws, a piece of wood to the back wall to hold up the break. Who is ever going to see it ?
Remove the shelf and fix it. You could use epoxy or replace the wood itself. Either way pull the shelf out.
That horizontal slat looks like it has a perfectly straight break, almost as if it was two boards and only the veneer was holding them together.
Remove the entire cabinet from the wall and replace those slats with pieces of real wood. Feel free to replace the stapled fiberboard shelves with wood and screws, while you’re at it.
Get used to it. Build quality is shit. I basically rebuilt the pantry with 1x2s and screws. Not pretty but nobody sees it but us, and it will not plague me again. When something breaks I fix it better than it was so I won’t have to do it again.
You could Use a piece of 1×2 cut to length and stain as close as you can, glue and screw underneath to support the broken piece. Maybe do this in all cabinets to match and add support
I’d just cut a piece of wood to length between the break and the floor and glue+staple it in place.
It’ll be a more secure repair than the original work.
I hate it when the Woood™ fails…
Source: Broken shelf in pantry.