To send a direct message to an event attendee, you can swipe left from anywhere to reveal your Messages screen.
Here you will see any new messages (noted by a number next to the person's name who sent you a direct message).
To send a new message tap the plus sign next to Direct Messages, which will present a list of people you can message.
Here you are given the option to send photos, videos, audio, your location, or a gif to your connections.
Note: depending on your events settings, you may only be allowed to message those who you're connected. To connect to an individual, find their profile and tap the Connect button. Once they accept your request you will be able to see all of their contact info.