How to repair a bulged exterior door threshold?
Step 1: Properly size the support lumber
What we see there looks like more than just swelling to me. It seems like someone jammed a full-thickness two-by in there out of haste. Swelling was just the final straw that caused rubbing.
You'll need to remove it. Use a reciprocating saw (Sawzall) to cut any fasteners running through it, then pry it out.
Then flex the threshold back into shape. Support the ends when you do--you don't want to tear it from the jamb sides.
Use a dry piece of lumber to fill the void. Cut it to the correct size (thickness and length), and use construction glue or silicone caulk to set it in place. If there's room you could toenail a few screws from the front.
Step 2: Flash the area properly
None of that wood should be visible. Fit a piece of aluminum stock or other water-repellent material under the threshold and down the face of the wall. Tuck it back under the bottom board if possible, and caulk that seam, or leave a short flange protruding outward from the wall and caulk under it. Caulk between the threshold and the flashing and along the side casing as well.
_________________________
_________________________|&lt;-- threshold
                      --*--
                         ||
            top board    ||&lt;-- flashing
            _____________||
                         ||
                         ||
          bottom board   ||
_________________________||___
                            *&lt;-- caulk beads (here and above)