To cast a video from your web browser, you should
-use https://app.put.io on Chrome, and
-make sure that your device and Chromecast are connected to the same network.
When these conditions are met, you should see a Chromecast option in the menu next to the video:
As long as your iOS device and Chromecast are connected to the same network, you'll see a Chromecast icon on the top right corner. Here's a gif that shows it all clearly: