I can see daylight through the door in the first picture, start with sealing that tight.

The corner seal is missing where daylight is visible. It is a vinyl covered foam wedge that should be on the lower jamb where you can see the residue of adhesive. I would add a nice full glass storm door to stop drafts and leaks.

@Kris would both bottom corners of the door need one?

Yes both sides should have one

As addressed in Freeman’s answer Poor door fitting against weatherstripping can be difficult to remedy. Also you should be sure the bottom of door is sealing well against threshold. Some thresholds are adjustable.

A storm door would help, but weatherstripping would be a better first step, much less expensive, and necessary either way.

Looking at the first picture, I don't see any way the water came through the door. It would have to have somehow defied gravity as it crept around the corner and then somehow got behind the paint. I'd look above the door on the outside for leaks. If you have water coming in above a door, it will tend to run down the side(s) of the door frame and cause the exact kind of damage you see there.

@JimmyJames I’ll take a look at that too.

@JimmyJames I took a look, and didn’t find anything that fits this description. It was a good idea though.

@JimmyJames The bottom of trim molding is usually exposed. A carpenter won't paint this for you unless you explicitly asked and it's in the quote. The water got in the house, sat on the floor, and was absorbed by the molding which is clearly evident by the swelling.

@MonkeyZeus There's no way a piece of dry wood would wick up that much water overnight. The OP also never mentioned water on the floor or floor damage. I've got a very similar situation on an exterior door and it is 100% due to water coming in from above the door on the outside. I don't know why you think this isn't possible.

@FrancisJohn After the storm, did you have water on the floor? Was that floor mat/rug wet? Is there floor damage?

@JimmyJames no, just the molding.

@JimmyJames Just because it's not in the question doesn't mean it's absent. It could have easily dried up by the time OP took the picture. Based on the swelling it is likely MDF anyways and will never recover to it's original size.

@MonkeyZeus there is also a chance that this has been a gradual problem that I’ve never noticed before. After this heavy rain was the first time I noticed it.

@MonkeyZeus Right which is completely consistent with water behind it. Carpenters often don't seal the back of trim either.

@JimmyJames I just uploaded two more pics. One of the entire front trim and the other closeup of the small ledge above the door. I’d say the seals look good enough at the top and at the sides, and the actual threshold seems setback enough. Not that I’m a professional, but I think replacing the weatherstripping and adding the bottom corner foam wedges with sealant should be enough here I hope.

@MonkeyZeus So the theory is that there was enough water on the floor for the door trim to wick it up but all of that water was dried from the rug and floor overnight but only the door molding was absorbing water, not the adjacent molding?

@FrancisJohn I hope you are right. As MonkeyZeus point out, that trim is fubared now and needs to be replaced. If/when you pull it down, I would inspect it and the cavity behind it for signs of water. I've seen water run straight through a door frame from above and out the ceiling below. You could have something like this as a result of a roof leak.

@JimmyJames That's a question partially for OP; they'll have to confirm/deny additional damage. Floor molding tends to be at least primed on 3 sides; ditto for the shoe molding. Either of these could be solid wood instead of MDF so the swelling would be less prominent. Additionally, the shoe molding would act a first line of defense for the floor molding. If the water was coming in from the back side then it would have swollen the floor molding due to the unprimed 4th face, no?

@MonkeyZeus "then it would have swollen the floor molding, no?" Not necessarily. When a door frame is installed, I'm sure you know there's a gap between the studs and the frame that needs to be shimmed. That can be a conduit for water that probably terminates at the bottom of the door frame. I doubt this happened overnight as well.

@JimmyJames the most prominent damage was the vertical trim on the interior side of the door. If I had to guess, I think it water was hitting the exterior front of the door, rolling down, and pooling in the corners. Due to the lack of the wedge backing and proper sealant, the weatherstripping and rain protection was comprised in the corners and made its way into the house at the bottom corners of the frame. From there it was absorbed by the interior trim.

@JimmyJames Beats me. I doubt OP is looking to turn this into a multi-day carpentry/painting headache unless enough evidence suggests that they should. FrancisJohn, if you have an unfinished basement beneath your entrance then check for signs of water infiltration on your rim joist, floor joists, and sill plate.

@MonkeyZeus unfortunately I wouldn’t have access to that portion of the basement, it’s mostly finished. But I haven’t seen any water damage there.

@MonkeyZeus Yeah. I would keep an eye on the spots that were wet. If this fix doesn't completely resolve that, it might be worth pulling the molding down for a look. It's not too much work and if you are careful when taking it off, it should go back on without much repair.

I also noticed the door bottom had some weathering/pieces of the front edge flap missing too. I just switched that out too, and just in time because the rain has begun!

The aluminum metal threshold that's outside the door itself may be pitched the wrong way, or may not have enough pitch for the water to run off to the outside.