It works through the app. You can only start downloading games you just purchased, not ones already in your library.
The mobile version of the Steam Community website also allows you to manage downloads, so if you visit your profile from within the app, you can view your games list and initiate a download.
I've always found it interesting that the mobile version of the Steam Community website lists the correct amount of games you own yet the desktop flavour also includes DLC and the like.