We don't have an official app for Android, at least not yet. ✌️
put.io works great in your device's browser, and you can always add it to your home screen as a shortcut.
Still, if you'd rather use an app, here's a 3rd party one we love and use:
Client for put.io from Vego Labs
This app, from Sri Lanka, lets you do most of the things that you can do with put.io, including downloading your files, adding new transfers and casting to Chromecast. You can find it here.