My name's Kel.
This looks like a newer home.
Do you have a warranty either in the purchase agreement or in state law that would require the builder to do the repair?
Have you spoken with your insurance agent to see if any of the damage is covered by your homeowner's policy?
No good news.
given the severity of the problem you'll have to remove the door.
I'm guessing given the damage there framing pan placed over the framing before the door was installed.
These can purchased as a piece for a particular door or a pliable membrane like Ice and Water can be put down.
Also guessing the flooring -- either a deck or concrete walkway are back pitched and letting rain flow back toward the house.
Inadequate caulk might have been used before the door was set.
Is there a drip edge over the door? Not one of those minimal built-in types, bu an actual drip edge. Even if there's a drip edge it may have been installed incorrectly. It may be letting A LOT of water leak in at the ends. If there's not adequate clearance between the siding the drip edge water can be drawn up into the wall.
Need at least 3/8" clearance.
Inspect the caulking along the sides of the door. There may be slight gaps that need to be filled. This is especially possible if the exterior's finished with stucco.
Have you gotten down on your hands and knees during a heavy rain to see if any water is getting by the glass seals and leaking into the underlying space?
Is there a gutter above the door that may be letting water into the eave? Any signs of water running through eave and into the house?
Any signs of leakage above the door on the inside -- especially brownish stains on the surface of the drywall?
If you're looking for a temporary repair -- caulk the ends of the drip edge that goes across the top. The ends where the siding wraps around the drip edge.
Caulk the sides of the door.
Run a bead of caulk along the bottom between door threshold and decking.
BTW. There's rarely a smoking gun in these situations. In other words a large obvious hole that's admitting water. It's usually a whole bunch of smaller breaks, fissures and caulking failures that cause the problem.
Does this make sense?
Have I answered your question?