What is the way to download the SMS messages from my iPhone to a Windows machine? Is there some apple cloud service that I can access for SMS messages?
SMS is a strictly cellular service, meaning all text messages use cellular connection. There is no way to access your text messages from the Internet(or by any other means that does not use the phone where he text messages are located). If you could clarify what purpose you need these messages, we could help find a solution.
EDIT: After clarification, it seems you would like the raw text content. If you have a Macintosh computer running OS X 10.10 "Yosemite" or higher and your iPhone is running iOS 8 or higher, you can use continuity to sync all SMS messages with your computer, then allowing you to manually copy/paste all of the content.
For a guide on how to set this up, see Apple's guide on Continuity