If you're using the Facebook app on your phone, there are a few different ways to access your message history. The first is to simply scroll down to Message History and select Load More Messages. This will load more of your past conversations into the app so you can view them.
Another way to access your message history is to go into the Settings menu and select Account Settings. From there, select General and then Manage Your Messages. This will take you to a page where you can view all of your past messages, as well as search for specific conversations.
Finally, if you want to export your entire message history, you can do so by going into the Settings menu and selecting Download a copy of your Facebook data. This will allow you to download a ZIP file containing all of your data, including messages.