Hi Abby
Very difficult to say. The soil looks like it has moisture but do check it by digging down a little and seeing if the soil was too dry. The leaves are rolled around which might indicate water or chemical stress. Those on the bottom are a bit brown which does indicate scorching. Was it too close to a heater? A closer picture would help make a better diagnosis.
Try watering it. Also cut of most of the dead leaves so the plant doesnt have to invest precious resources into those sickly leaves. It might bounce back. Put it on the sill of a south facing window and you might get some new growth.
Have you other cayenne pepper plants? What about your other houseplants?

Dont get discouraged and keep trying just imaging how nice the peppers will be in the end

UPDATE
Thanks for the close up. I would say that this plant died becuase of disease and Cayenne peppers suffer in nurseries from Verticillum and Fusarium. These are both fungi. There is also a bacterial wilt but that usually has spots on the leaf.

You should throw this plant and the soil in a sealed bag and then into the garbage. Not on the compost if you have one. Wash the pot and you can re-use.