Remove the salmon from the fridge and let it rest for 15 minutes before cooking. This will allow the salmon to come to room temperature. If you add cold salmon to a hot pan, it will seize up and will not be as tender and juicy. Also make sure the salmon filets are patted dry.
Preheat the oven to 425 degrees
Preheat the cast iron skillet on medium high and add olive oil. Season the salmon filets with the salt and pepper while the oil heats. The oil should be hot (shimmery) when you add the salmon, but not smoky.
Add the salmon to the pan and sear skin, side down, for 4-5 minutes without moving the salmon so that it properly sears.
Remove the pan from the burner and put in the oven for 5-6 minutes (for medium rare) until the salmon is just opaque in the center. Time in the oven will depend on the desired doneness (medium rare, medium well, etc) and the thickness of the filet. The salmon should be opaque and flaky.
Transfer the salmon to a plate and cover with foil to keep warm.