Assuming you have already logged into phpBB. Click on the “0 new messages“  link as shown below.

Click on the private messages tab, you will see a screen like this below.

Enter the name of the user we are messaging, or use the find a member tool to locate them, after you are done with that click add button,
Enter the subject of the message under the subject line and ofcourse the body of the message and then click submit button.
You can also check the messages in your inbox, outbox and sent messages from here.

You now know how to send a private message in phpBB.