Based on the picture and what you described, you have tile set over vinyl sheet flooring that was put down over 1/4 in plywood used as underlayment, and the gray stuff underneath is the old floor boards. I'm assuming you want to tile this because it's no longer going to have something sitting on it and you just want it to match for the next few years until you renovate the whole floor.
The easiest quickest thing to do would be to fill the gap in the thin plywood underlayment with more plywood, either same thickness or a little thicker to make up for thickness of vinyl, then set tile over that adhering it with thinset directly to that plywood. If you don't put down plywood, you'll need to have the thinset a good deal thicker which means a) more thinset to buy+mix+handle, b) sloppy deep mess of thinset to work with and level tiles in, c) maybe over thickness spec for the thinset. Also you'll be adhering directly to the floor boards and that's not great because a) you'll probably ruin those boards and will have to replace them if you do anything else with the floor later and b) they're almost certainly not a good substrate for tile because they flex from load and environmental changes.
Quarter inch plywood over floor boards is still not great, that tile is likely to crack either when you set something heavy on it, when the seasons change, or just from walking around in the kitchen. Ideally, you'd make sure the structure underneath that floor is sufficiently strong so the floor doesn't flex, tare up the whole floor down to the floor boards, put down proper thickness of plywood, maybe some kind of waterproof membrane if you want a waterproof floor, then tile the whole thing wall to wall.
But as a temp fix, you can put down more plywood to make the hole less deep and protect the floorboards, set tile on it with thinset, and grout it. You can try to knock out the cut tiles like Jack said if you want to make it more continuous. For more info on setting tile, invest a few bucks in this book from amazon and check out the John Bridge forums.