Windows Live Messenger provides options for automatically maintaining the list of conversations. But if you do want to store the conversation then you can change it using the Messenger Options menu.
Click on the Tools –> Options menu –> Navigate to Messages tab on the left hand side.
Under Message History, un tick the check box with label as Automatically keep a history of my conversations. And click Apply and then OK button to confirm and Save the changes. This would clear the message history in Windows Live Messenger.