"Liberty", as you might expect, had a broad range of meanings then just like it does now, but given the context you mention it's presumably intended as a reference to the idea of liberty as a uniquely developed characteristic of Britain's political system. In the period you're referring to, the idea of prosperity -- symbolised by beef among other things -- and liberty as being unique properties of Britain was very common in patriot literature.

To understand the specific significance of liberty, you need to go back to the wider context of the "Whiggish" understanding of the British constitution that became dominant from the 17th and 18th centuries. This saw it as a peculiarly well-balanced political order, approximating the ideal of the classical republic with its mixed constitution (House of Commons, House of Lords, monarchy in this case representing the classical balance of democracy, aristocracy and monarchy).

From the early to mid 18th century in particular, however, this was increasingly situated in the context of a narrative that saw English liberties and the institutions of constitutional monarchy as having been handed down from time immemorial from the ancient Anglo-Saxons, and successively reasserted from the Magna Carta to the Glorious Revolution.

When asserted in this way as an aspect of English genius, this kind of patriotic narrative made for a sharp distinction between Britain, and especially the English people, and the other European nations, which were generally seen as in hock to various kinds of despotism. During the French Revolution, there were some British radicals who supported French aspirations, but, particularly as British opposition hardened and the Revolution became more radical, many of a patriotic bent reasserted the distinctiveness of British liberty in contrast to French liberty. A good example of this is this famous caricature from 1792, which summarises the narrative: the long tradition of ordered British liberty on the one hand, and the sudden interruption of French violence on the other. (Note the Magna Carta being held by Britannia on the left.) This is the kind of distinctive idea of "liberty" that they are alluding to in your quote.

Some sources: John Dinwiddy, 'England', in Dann and Dinwiddy (eds.) Nationalism in the Age of the French Revolution; Lee Ward, 'Eighteenth-Century British Constitutionalism' in The Politics of Liberty in England and Revolutionary America.