To fix this, you simply need to add this line into the AndroidManifest.xml under Assets/Plugins/Android, setting the targetSDKversion to a value lower than 23 (e.g. set it to 21 or 22):
<uses-sdk android:minSdkVersion="17" android:targetSdkVersion="22" />
This will make it work on any device (including those on Android 6.0, as well as the ones on older versions, down to the minSDKVersion you have set)