How to activate TheTruthSpy features without touching the target phone?

You can do this via the website http://my.thetruthspy.com or use SMS commands.

On the website, login to your account > open page Cell info > adjust TheTruthSpy parameters as you want > Save. The new settings will be applied to TheTruthSpy on the target phone in the next sync.

With SMS commands: you need to set a monitor number first, then use this monitor number to send SMS commands to the target phone number. Then, based on your need, you can choose suitable command in the List of SMS remote control commands part of the user guide document...