If this does not mean the inherited actual sin of Adam (and with this the Christian East totally disagres), what is it?
You like the easy questions with nice, short answers, eh?
The short answer is that it means that the guilt of Original Sin is manifested by a lack of Original Justice (Grace), as Adam's guilt meant he lost Grace, and we've inherited that same privation even without being personally responsible for it. If you want a more detailed explaination, brave the flood below, which even I admit is simply too long. As the saying goes, sorry for the long letter; if I'd had the time I'd have written a shorter one. :p
Confusion comes, I think, from the unfortunate use of the term "guilt of Original Sin". While the use of the term "stain" is present in both East and West, the use of the term "guilt" to describe Original Sin is not, and it doesn't always call up the image of what's actually being described.
In Western usage, the "guilt of Original Sin" as manifested in Adam and Eve was a privation, or seperation, from the Grace of God, and the resulting disorder in him due to this. Sin is not "something", it's has no form of its own; rather it represents a stepping away from God. In being personally guilty of sin, Adam deprived himself of God's Grace that was in him by original design (called by St. Thomas Aquinas "Original Justice").
Therefore the "observable" manifestation of his sin is the absence of Grace, and the resulting disorder; even though it's not a thing, it can be recognized, just like the void of space is recognizable even though it's not really "there" by its very definition. It's really at its heart an apophatic recognition in the most extreme sense. To put it another way, how can we tell that Adam had sinned? He lost his Grace. So we say that his guilt, the "stain of his sin", is the privation of Grace.
Now we can't help but notice that his descendants inherited this same privation, even through no fault of their own. We say they "inherited his guilt" not because we somehow have to make recompense for something "we" did (although one can use such collective terminology to describe things), but because the manifestation of Adam's guilt is present in us as well by virtue of our descent from him.
Of course there have been various Latin theologians who have tried to tackle the question of "why" and "how" this privation would be passed on, and some took some rather odd and questionable stances, but the fact remains that it's a privation in their view. There's also the fact that Latins due often speak of atonement through Sacrifice, but I think that is easily explained in the Scriptures themselves, and the image of Jesus as the Lamb. Most confusing is the fact that Latin theologians would juggle the apophatic description of Original Sin at the very same time as speaking of it as if it were a "something". This is especially visable with St. Thomas Aquinas, which I'll show in a moment.
All of that is saying that Original Sin is spoken of, and understood, in much of Latin thought as simultaneously being "something" materially speaking, as we can see the results of it, and indeed it does have a material shape in the case of physical death for example, while at the same time it is ultimately nothing but the lack of God's Grace. This is why St. Thomas Aquinas could say with a straight face, "Original Sin is concupiscence, materially, but privation of Original Justice, formally."
Unfortunately, the term "guilt" tends to give the impression of something tangible, when it's actually a lot more complicated than that. Even worse, it gives the image of personal responsibility.
As for why such a statement was deemed necessary, I think it has a heck of a lot more to do with the Protestant West than the orthodox (including Orthodox) East, but this post is long enough already :p
Peace and God bless!
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