What does Sauron mean by 'The East will fall...So shall the Kingdom of Angmar rise'?

The East in this context presumably means the north-east of the civilised portion of Middle-earth, where the Hobbit movies largely take place — Thranduil’s kingdom in Mirkwood, Laketown, Dale and Erebor, and perhaps Dain’s dwarven kingdom in the Iron Hills. Angmar is well to the west of all of those.

Another option is that “the East” simply means all of Middle Earth, in contrast to the western lands of Valinor (unreachable if you’re not an elf, with only one or two very special exceptions) and the drowned lands of Numenor and Beleriand (which have thus already fallen).

While I think Mike Scott makes an interesting point I'm not sure he's hit the nail quite on the head. While the North East is indeed where the story of The Hobbit takes place I think for the Necromancer, and in terms of the Kingdom of Angmar, "The East" is more likely to refer to the Eastern kingdom of Men, Gondor.

As I've explored in another answer about the Witch-king's motivations the motivations of the stronghold Angmar was to destroy the Kingdom(s) of Men in the North West, which they had managed successfully. Naturally the progression from there would be to destroy the other (Eastern) kingdom too. Sauron (the Necromancer) had a personal hatred for the Men of Gondor (and previously Arnor) as they were the 'faithful' of Númenor, who opposed his control of Ar-Pharazon and his attempts to make war on the Valar. It is possible the Necromancer referred to all the free people's living East of the Misty Mountains in Rhovanion in addition to Gondor.

As I have just come to realise in my comment, the East could mean Endor in it's entirety, where the West means Aman, the Blessed Realm and the East meant Endor, Middle-earth. This seems possible as Sauron's idea was to dominate all of Middle-earth.