Replacing exterior door sill - making it fit
The rubber thing is called a door sweep and they come in many styles and materials. It's purpose is to seal off the bottom of the door from drafts. The swing orientation and threshold style usually dictate the sweep. Some mount under the door, on the front side and the back side. Bottom sweeps are often a softish D shape with the crest pointing down. Fronts and backs are often a bristle or rubber blade.

Undercutting the door is an option. If it's an exterior wood door undercutting is easy.

Also relocateing the threshold towards inside or outside may help your situation. If you are able to move the threshold to the exterior and put the sweep on the exterior of the door, the door will not need to close over the threshold.