The left side of the door sweep seems to be tight against the door, while the right side, closest to the camera, appears to have a significant gap. The screws closest to the camera don't seem to be quite tight enough. Of course, water doesn't care and will get in through both capillary action is not your friend in this situation.
You're probably right that water is getting into unprotected wood through the screw holes and that some silicone caulk would do the trick. Undo each screw, squirt a smidge of clear exterior grade silicone in the hole and reseat the screw. Additionally, refinishing the door completely is probably the only way to make the appearance of the damage go away. (I'd suggest a visit to the Woodworking sister site to get all the info you could ever want on the how-tos of finishing a door.)
Once you've got the short term fix of some silicone in place, a couple of long-term fixes:
- Put a storm door of some sort in front of this nice wooden door. That would significantly reduce the amount of rain getting onto the door in the first place. The door appears to be fairly simple, so a standard screen/storm door might be appropriate. If you prefer the look, a full-glass storm door would do well to allow the wood door to be more visible with only a thinish metal frame around the outside.
- An overhang of some sort would help too. That would have the added bonus of giving you and/or your guests someplace sheltered to stand while waiting for the door to open.