Does the other person know that you have blocked them in xbox live?

The person you have blocked has no idea that they are blocked. They can still send you a message/invite but you will never receive it. The original sender gets no message or indication that they are blocked.

You can clearly see that you have been blocked on Xbox.com. Log in through that website and try sending that person an Add Friend submission. It will clearly state if you have been blocked.

For example, I'm seeing the exact message as I have been blocked by someone. So, if you have been blocked by another user, you will see it. On the Xbox One, you can't see any indications of it.