Your screenshot is so small and doesn't show us anything about your composition so we have no idea how you are doing what you are doing. There is also no comparison between what you are seeing in the composition window and the rendered output.
Please give us a screenshot of your entire comp with the problematic layer selected and all changed properties revealed. Just press the u key twice and then capture the screen and upload the full sized image.
The layers highlighted are the ones with bicubic interpolation enabled. It looks fine in the preview window and in premiere, since this whole after effects comp is a dynamicly linked comp between AE and Premiere. The problem is, when I render this comp in After Effects, those highlighted layers become blurry. Also, if you're wondering why I'm rendering this comp even though it's dynamically linked with premiere, it's because I do it to reduce time in rendering when I color correct/grade in premiere.
In your screenshot, there are no layers in your composition with bicubic interpolation enabled. You have some with Draft quality and some with Best/Bilinear quality, but none with Best/Bicubic.