From the homepage, you can see the number of outbound message sent in the current month as well as the total number of your subscribers.
From the Communication & Marketing menu, you can click Messages, get to the Messages side menu.
On the side menu you can click to view the Outbox to see how many messages were sent when. The number in the sent column shows how many people a specific message went to.
Scheduled shows messages that are scheduled to be sent, and the date and time they will be sent.
The Inbox will show text messages received through your keywords.
Click on a keyword to show the messages. This is useful for Events, where donors text more than just the keyword, they also text their pledge and name, etc.
You can also see how many messages have been sent in a given campaign by going to Reporting > Campaigns.