Note: You probably routinely delete your texts individually so you no longer see them on your phone. Did you know this does not delete from history? They are all still there!
To see your current usage before you clean up: Go to Settings, General, Manage Storage, and select your Messages app. You might be surprised at the size.
Here's the current storage. My usage is small since I'm only saving 30 days now.
Now let's proceed to delete... Go here to Settings, Messages and select Messages.
In messages, scroll down until you find Keep Messages and select it.
In Keep Messages, select your choice. When you change it from Forever, all messages older than your choice will be deleted from history now, and, evidently, ones on your phone, too.