How to fix this broken floor
The floor joists are rotted in that particular area. It could be the same all over. The only thing you could try is to remove some more of the decking and "sister" the joists (Nail a second joist alongside the old one). This will give you some new area to nail your decking. You might be looking at replacing this deck in the near future. If all of the joists look like that, someone could get hurt.