If a number of things are F, then they are F in virtue of something (i.e., something that makes them F as opposed to those things that are not F – this is a simple semantical claim, not involving serious commitments in ontology.)
This claim does not entail that there is some single thing in virtue of which all things that are F are F. (Trying to draw this conclusion would involve the quantifier-shift fallacy.)
This claim does follow, however, if we can show that if a number of things are F, then they are F in virtue of one and the same thing. (Anselm tries to show precisely this, concerning goodness, in c. 1, p. 12).
Now we can show this by pointing out that since being F is common to all these things, they have to have this common feature in virtue of something that is common to them all.
But anything that is common to many things has to be some one thing equally related to these many things.
So, we can conclude that
(I) If a number of things are F, then they are F in virtue of some single thing, which is the same for all.
(This is the strong Platonic-sounding principle Anselm starts out with. However, that it is not simply a Platonic principle will be clear from the immediately following considerations.)
The single thing in virtue of which all things that are F are F cannot be intrinsic to these things in the way their own F-nesses are. For if we call the common thing 'F-ness', and the intrinsic F-ness of a thing a 'the F-ness of a', and the intrinsic F-ness of b 'the F-ness of b', then this assumption would entail that the F-ness of a and the F-ness of b could not independently come to be and cease to be, which is obviously false. This can be seen if we consider that for an F-ness to be intrinsic to an x means that for that F-ness to be is for x to be F, that is, that the act of being of that F-ness is nothing but the act of being of the F-ness of x. But then, if F-ness is intrinsic to both a and b, this means that the F-ness of a and the F-ness of b have the same act of being, that is, the one cannot cease or come to be without the other, which is false. (This is just another way of stating the Boethian argument I reconstructed in my Stanford Encyclopedia article on “The Medieval Problem of Universals”.)
But if that common thing cannot be intrinsic, it has to be extrinsic. So the relation indicated by the phrase “in virtue of” or “through” [used in the translation of Anselm’s text] in “that in virtue of/through which all things that are F are F” should indicate an extrinsic causal relation, namely, efficient causality.
But the extrinsic universal cause in virtue of which all things that are F are F cannot be F in the same sense in which the things that are F are F. The simple reason for this is that if it were, then it would have to be a cause of itself in the same way as it is the cause all these things, but this is impossible, since nothing can be an extrinsic efficient cause of itself. So, it has to have some universal power, which does not make it F, but which is able to make all F things F, whence we can say that this cause contains the F-ness of all F in its power. It is precisely for this reason that we can say that although this cause contains F-ness in its power, it is not just an F, it is F-ness itself, or rather it is even above F-ness. (This is the point illustrated by Anselm in his "brightness-analogy", in c. 6, and analyzed in more detail in cc. 15-16, and 26-28.)
On the basis of these considerations and (I) above we can quite plausibly reconstruct Anselm’s main argument as follows:
(II) If a number of things are F, then there is a single universal cause in virtue of which they are all F.
(III) But all things that exist are good (to some extent, at least) and all the things that there are exist.
(IV) Therefore, there is a single universal cause in virtue of which all things are good and existent.
(V) The universal cause is goodness itself, and being itself, or indeed, it is above goodness (as we know it) and above existence (as we know it).
(VI) But then, since goodness and being itself cannot be lacking in any perfection, and cannot be dependent on anything for its being, it cannot be restricted by space and time by spatio-temporal boundaries, and it cannot be meted out in space and time along some spatio-temporal dimensions.
(VII) Indeed, it has to be absolutely simple, i.e., it cannot have any parts, for whatever has parts is dependent for its existence on its parts.
(VIII) Therefore, it is absolutely one, for, since a thing is one insofar as it is an undivided being, only that thing can be absolutely one, which is absolutely indivisible.
(IX) But then, it cannot have accidents (for accidents make a composite with their subject), but all properties it has belong to it in virtue of its essence, which is nothing but itself, and so it cannot change, to lose what it has or to acquire what does not, for it has all perfections essentially united in its single, undivided, unlimited act of being.
(X) But a being like this can only be God; so God exists.
For now, I’d just like to call your attention to the important difference between this long chain of reasoning and the argument for God’s existence in the Proslogion. What difference do you think it makes that here we conclude the existence of God only after first proving the existence of a universal efficient cause and then gradually “loading” its concept, until we conclude that it has to be God, as opposed to first saying what we should understand by the name, and then directly concluding that, given this understanding, there must be something in reality corresponding to the name?