I play Trials almost exclusively on Android, so I can help you out here.
Go to fenoxo.com in your mobile browser.
There should be a small dropdown menu under the FENOXO'S BLOG title.
Click on the dropdown menu and press "PLAY".
You will see the games hosted on the website (TiTS, CoC 1&2, FoE). Under Trials and Tainted Space, click on "download for ARM Android".
The download will begin. It's not particularly big.
Once it's finished, tap on it from the notification bar. You will be blocked from installing it.
To fix the previous step, go into your Android settings > security and lock screen.
Enable "allow installation from unknown sources".
If TiTS is no longer in your notification bar, then go into your file manager > downloads and you will see the TiTS apk you just downloaded. Tap it and install it.
Once it's installed, delete the apk. You no longer need it and it only serves as clutter.
You should now have an icon on your home screen or apps page called Trials in Tainted Space. There you go.
To update the game when a public update is available:
Go to fenoxo.com again, and go to "PLAY".
Download ARM Trials in Tainted Space again.
Once it's done, open it. You will be promoted to update an existing app (TiTS).
Click yes.
When it's done installing, delete the apk.