If you reply to another message by clicking "Reply to Quote", the message you are replying to will appear in the white box within your message so others reading it will know what you are replying to.
By using this feature, it will also keep all replies made to that post available in one place and you can readd all of those replies by clicking "Read Replies to this message". Here's an example, I'll make up some names:
Lucy posts a message, Robin clicks the reply with quote and replies to Lucy's message. If Lucy wants to see if anyone has replied to her, all she needs to do is click the Read Replies to this message and she'll see Robin's reply.
If Mary, Ann, Kate and Rhonda all reply with quote to Lucy's message, then Lucy will see all of their replies as well when she clicks that link.
If you reply without using this feature (using the post message button instead), your reply will not be added to Lucy's list of replies and she may not see it.
Hope this makes some sense!