How can I repair a big gash in my kitchen floor?
You cannot "fix" it without replacing the floorboard, but you can patch it to look close.

That looks like IKEA's pressboard laminate - that's not wood you're seeing, that's paper.  You can check their stock for some extra boards if there's one near by - sometimes they might have some extras in returns.

So - get yourself some dark stained wood filler and a 2" putty knife, and a small "wood stain marker" that matches the color more exactly.  You can find all of these in among the paint aisles in any big box hardware store.  You'll also want some 100 grit and 220 grit sandpaper.

Fill the hole with filler and smooth with the knife.  Let it dry and sand smooth first with the 100 grit then the 220 grit.

Finally, stain the filler to match, and let dry.