First you need to make a challenge, (by clicking the CHALLENGE button on the upper right), and then you should be able to upload a level to the steam workshop!

Thanks to this great Q and awesome responses I wrote up a guide on how to upload both Challenges and Endless levels to the Steam Workshop from the level editor: https://support.doborog.com/clone-drone/uploading-levels-to-the-steam-workshop

Awesome to see you all sharing so many levels!