Here's a parable to understand Anscombe's point about the 'moral ought'.
Imagine a young woman in school who devotes herself especially to music (she plays the piano), gymastics, and mathematics. Each of these areas, clearly, has its own standards of excellence. What is 'required', what is to be done, is one thing in gymnatics, another thing in music, and something else (the demonstrandum) in mathematics. It would be absurd to suggest that this person in pursuing her education is measuring her actions against some common notion of 'requirement'. It wouldn't help for her to keep constantly in mind that "I'll aim to do always what is required."
Now suppose that her father, whom she loves and admires very much, lately takes a special interest in her education and tells her something like, "It would please me greatly if you excelled in all areas of your study." Perhaps he is a perfectionist, and the woman gets the sense that in meeting the standards distinctive of each of her areas of endeavor, she would also meeting be her father's standards for her. Now it would indeed be possible for her to conceive of her studies with a certain unity, as aiming to 'please her Dad' or 'meet his expectations (or standards)'. But note that this unity would be a posterior, and there would be no question--it would be absurd to suggest--that the father's interest in her education creates standards in mathematics, music, or athleticism.
Anscombe believes that historically Christianity allowed for an analogous unifying of the moral life. It is correct, she believes, to think of requirements ('what is to be done', officia) as, initially, varied and peculiar to different domains of action--what is to be done by a soldier in battle is different from what is to be done by a merchant exchanging goods in the marketplace, etc. But if one holds that the requirements of practical reason in these various domains are expressions of a 'natural law' which is promulgated, as it were, in the very nature of the practical reason that we have, then it becomes possible to view what is required under any virtue as falling under the concept of 'required by divine law'.
In Anscombe's view, when moral philosophers of the 1950s (e.g. Hare) were aiming to analyse 'the concept of morality' and looking for the distinctive 'logic' of the word 'ought' as, it was supposed, used in a distinctively moral sense, they were engaged in a fundamentally incoherent enterprise, that came of imagining there to be a unity to requirement in action, which could not intelligibly be accounted for except for the postulation of a divine legislator or the equivalent.
One finds a slightly similar line of thought in Aristotle's Ethics (although Anscombe does not mention this in her article). I mean Aristotle's discussion of 'general justice' in V.1. He admits that there is a sense in which what is required under any virtue may be regarded as required in the same way, if we regard the acts of the various virtue as all obligatory under some system of law (hence an act of courage and of commercial justice are both alike in being 'what's law-abiding', nomimon.) (I would add that Anscombe's criticism of the concept of 'moral ought' has a similar spirit to Aristotle's criticism of a uniform notion of 'good' in NE I.6.)
It's clear that Anscombe's view, and Aristotle's, have nothing whatsoever to do with the view known as 'voluntarism', viz. that moral obligation must ultimately be explained by appeal to the will of a legislator.
In fact, Anscombe formulates her view in a way that makes it very clear that she rejects voluntarism:
To have a law conception of ethics is to hold that what is needed for conformity with the virtues failure in which is the mark of being bad qua man (and not merely, say, qua craftsman or logician)--that what is needed for this, is required by divine law. Naturally it is not possible to have such a conception unless you believe in God as a law‑giver; like Jews, Stoics, and Christians. But if such a conception is dominant for many centuries, and then is given up, it is a natural result that the concepts of "obligation,” of being bound or required as by a law, should remain though they had lost their root; and if the word "ought" has become invested in certain contexts with the sense of "obligation," it too will remain to be spoken with a special emphasis and special feeling in these contexts.
In her phrase, 'what is needed for conformity with the virtues, failure of which is the mark of being bad qua man', she acknowledges standards of action which provide, as it were, the reasons why the divine law is as it is.
Anscombe of course knew Aquinas well enough to recognize that this is how he conceived of the natural law also! ("Non enim Deus a nobis offenditur nisi ex eo quod contra nostrum bonum agimus", ScG III. 122, and like principles, were surely known to her.)