The checkmarks act as a "read receipt," a way to tell if a message has been seen or not!
Two checkmarks appear next to the recipient's name when they have seen your message either from their email OR from their Acceptd account. Recipients without checkmarks have not seen your message yet.
Hovering over the checkmarks reveals the date and time stamp of when the recipient last saw your message.
If you respond to a message, the checkmarks will disappear upon refreshing your page. The checkmarks will reappear when the recipient has seen the new message.