Websites can only 'recognise' you if you've allowed them to place cookies on your computer in order for them to do so. Your post suggests that your browser isn't accepting Hotmail's cookies. Check your security settings; they're probably too high.
Did you tick the 'Remember me on this computer'/'Remember me' option when logging in? If not you should. This will remember your password and you don't have to type it in again and again each time you log in.