Kitchen floor rotted.
Depending upon when your home was built, you should have two layers under the vinyl flooring, the flooring and the subflooring.
You will need a circular saw with an old blade (you WILL hit nails), a crowbar, flat bar and claw hammer.
As jproffer said, set the blade depth so that it only cuts through flooring and not into any joists.
Cut down the middle of the nearest joists outside the rotted area, and a straight line between joists. It's usually easier to remove no more than a couple of square feet at a time, if the rotted area is large.
Install 2X crosspieces between the joists, halfway under the old, good subflooring and half into the removed open area for gluing and nailing or screwing to in addition tho the main joists.
I always use pressure-treated plywood when replacing rotted flooring anywhere that there is water likely to damage it again.