To apply an outfit style (which is what you're trying to do - it's a different thing from a costume), you need to use an outfit station, which are also used to dye armor. Go to the Outfits tab and make sure you have "Outfit 1" selected, and then you can apply any outfit styles and/or dyes you have available that you want to use.

Note that if you ever just want to dye the armor you're wearing, you'll want to have "No Outfit" selected so it doesn't charge you gold unnecessarily.