The feature works when you tap on any message in a conversation (be it text or media, group or one-on-one). A reply button will appear at the top of the screen. Tap on this button and the message you tapped on will appear as a preview inside the text entry box, which is where you can add your reply.
The reply will hit the screen with the original message on top of it. The name of the sender is printed in a color that matches his or her color used in group chats. Anyone involved in a WhatsApp chat can have their quote highlighted in this fashion, and those using "Quick Quote" can even use the new feature on their own quotes. Anyone who clicks on the attached reply will be sent to the part of the conversation where the original message was lifted from. The original message will be highlighted in blue for a brief time. This helps others understand the context of both the original quote and the reply.
It appears that the new Quick Quotes feature appears on the 2.16.118 beta of the messaging app. There are two ways to check it out now. You can become a beta tester by going to the Google Play Store and agreeing to test the latest WhatsApp beta version. Or, you can use APK Mirror to sideload the beta on your phone.