Let's get Moore's text up on screen :
I can prove now, for instance, that two human hands exist. How? By holding up my two hands, and saying, as I make a certain gesture with the right hand, 'Here is one hand, and adding, as I make a certain gesture with the left, 'and here is another'. And if, by doing this, I have proved ipso facto the existence of external things, you will all see that I can also do it now in numbers of other ways: there is no need to multiply examples. (G.E. Moore, 'Proof of an External World', Philosophical Papers (London: George Allen & Unwin, 1959), 144.)
The two hands are (assumed to be) examples of 'external things', hence of an external world in which they exist.
The shape of the argument is :
(1) Here are two hands.
(2) If hands exist, then there is an external world.
(3) So there is an external world.
There does appear to be (no, I'll say there is) a problem here. It can be spelt out as follows :
(1) already assumes the truth of (3). (1) lends no credibility to (3) since if (3) were not true, (1) would be false. Moore may have some epistemic justification for asserting (1) but no more justification than he has for asserting (3). So (1) makes (3) no more, or less, credible than if (3) were asserted on its own.
G.E. Moore, 'Proof of an External World', Philosophical Papers (London: George Allen & Unwin, 1959), 126-48.
Annalisa Coliva, 'The Paradox of Moore's Proof of an External World', The Philosophical Quarterly (1950-), Vol. 58, No. 231 (Apr., 2008), 234-243
James Pryor, 'What's Wrong with Moore's Argument?', Philosophical Issues, Vol. 14, Epistemology (2004), 349-378.