To use this feature, simply tap and hold a notification on the lock screen or in Notification Center for a split second. Once you feel haptic feedback from the Taptic Engine, let go and the notification will expand, providing more detailed information and contextual shortcuts, such as a reply field for iMessage conversations.
Haptic Touch is simply a marketing name for a long press combined with haptic feedback from the Taptic Engine. The feature is a substitute for 3D Touch, which Apple wasn't able to include in the iPhone XR, as it had to remove the pressure-sensitive layer from the screen to achieve a nearly edge-to-edge LCD.
Haptic Touch works in only a few places, such as the flashlight and camera shortcuts on the lock screen, and in Control Center to pop open additional toggles and menus, but Apple recently confirmed it is working to bring the 3D Touch replacement to more places across iOS over time, and notifications is a start.
Haptic Touch can only be implemented for actions that don't already rely on a long press. For example, long-pressing on an app icon on the home screen enables "wiggle mode," allowing apps to be deleted or rearranged on the home screen, so 3D Touch's Quick Action menus are not supported on the iPhone XR.
Haptic Touch also doesn't support Peek and Pop for previewing content such as websites and messages since the feature relies on detecting multiple levels of pressure, and the iPhone XR does not have pressure sensitivity at all.
On the iPhone 6 and earlier, which lack 3D Touch and Haptic Touch, a notification can be expanded by sliding it to the left and tapping View. This is still an option on the iPhone XR, but the Haptic Touch solution is more convenient.
Haptic Touch support for notifications on the iPhone XR is functional in the second beta of iOS 12.1.1, seeded to developers and users enrolled in Apple's Beta Software Program on Wednesday. The update is a relatively minor one overall, so it will likely be released to the public fairly soon.