I've googled this problem and still can't find the right fix. Hoping someone here has a better solution. I have an Iphone 5s. It says I have 6 gigs of Stored Messages. I've delete messages etc and still can't find a way to get rid of anything.
Go to the Messages app and hit Edit in the top left corner. Select entire threads and hit delete. That's seems like the most efficient way to me.
I think the only way to do this without an app is to change how long your phone stores them. Go to Settings; then Messages; then switch the setting from "Forever" to "30 Days." Otherwise you are stuck swiping them to delete or doing what HR suggested.
RasaqBoi... I think you might want to read this entire thread at the APPLE forums: https://discussions.apple.com/thread/5367022