How can I repair this rotten entry threshold?
Hard to confirm without pictures, but you might have a void below the wood where water gets in and is causing the wood to rot.
Remove the rubber gasket, then remove the metal threshold (if you have one). Next remove the rotted wood. You may have to cut away since the wood will run under the jamb.  Fill the void (if there is one) with quick setting concrete. Then just replace the wood. Use pressure treated wood to prevent rotting.
OR, just hire a repair/handyman.