You will have been logged into the Virgin Trains website at some point, which has then set a cookie. Cookies last often 90 days. Therefore subsequent visits can also identify you. Hence why some websites such as Facebook can keep you logged in for so long.

I'm sure BA will do similarly, I know for a fact there's a part of BAEC home that shows your recent searches, not sure if this includes when you're not logged in.
Budget airlines have been known to use cookies to proactively raise fares (e.g. they know that if you look once, and look again and see the fares have been raised, you'll impulse book).

Short answer is to clear cookies in your browser and this will clear the slate. There are various addons you can get for Google Chrome and otherwise which would allow you to selectively choose which sites you allow to set cookies and so on.