That's a pretty common perception among anarchists, and it's not without foundation: the Spanish anarchist movement immediately prior to and during the civil war was and remains the global high-water mark for anarchism as a mass movement. For a brief period during the early months of the civil war, certain areas of Spain — particularly in Catalonia — came closer to establishing a large-scale, functional anarchist society than at any other time or place in history.

The Spanish anarchist movement was dominated by the anarchist trade union, the Confederación Nacional del Trabajo (CNT) and an affiliated militant secret society, the Federación Anarquista Ibérica (FAI). All told, the CNT-FAI and affiliated anarchist groups claimed almost 2,000,000 members in 1936. That's a genuinely astonishing number — and it gave the anarchists a significant power base in republican Spain in the early days of the war.

Chronologically, you have to understand how critical the anarchists and revolutionary left were in preventing the military coup in July 1936 from succeeding outright: in Madrid and Barcelona, among others, the failure of the uprising to successfully seize control on 18-19 July was thanks in large part to spirited resistance by anarchist, socialist and communist groups.

The role of workers in the initial resistance was to become a major source of political capital for the revolutionary movements as the situation deteriorated into civil war, a social revolution engulfed the republican zone and a spectacularly uneasy coalition of socialists, communists, anarchists and republicans was formed.

There were anarchist ministers in Francisco Largo Caballero's national government — and not token appointments, either; at one point, anarchists controlled the ministries of justice, health and the economy. Similarly, there were anarchist ministers in Luis Companys' regional government in Catalonia.

For a brief moment, the social revolution in Spain appeared to many as though it may actually herald a fundamental reimagining of Spanish society. In rural areas, the anarchists carried out extensive collectivisation programmes; in industrial Spain, workers' committees (some anarchist, some socialist, some communist) took over the means of production.

The wave eventually broke in the spring and summer of 1937, however, when internal tensions in the republican zone boiled over during the May Days in Barcelona. The Soviet-backed Partido Comunista de España (PCE) toppled Largo Caballero's government, and took a dominant position in the socialist Juan Negrín's new cabinet.

Thereafter, the communists set about rapidly dismantling the anarchist and Trotskyist left's power case: rural collectives and workers' committees were largely dissolved, revolutionary militias were disarmed and integrated into the new Popular Army, and the anarchists were all but excluded from the national and regional governments. The CNT-FAI continued to be involved in the resistance against Franco — anarchist columns continued to fight on the front lines, and anarchist workers continued to organise and agitate on the home front — but their influence was largely broken, and almost all of the gains they had made in the first months of the war were lost.

It's a romantic and deeply tragic period in history for many anarchists; in popular memory, it's a time of resistance against fascism and of optimism in the new Spain they were trying to build behind the lines. But it's also a memory of promise and progress stolen from them by Stalinists — and it's a feat that hasn't been replicated on that scale before or since. So that, broadly, is why your local anarchists talk about it in such terms.

(For further reading, I'd recommend Hugh Thomas' The Spanish Civil War as an overview of the war as a whole, and Stanley Payne's The Spanish Revolution for an examination of the revolution's rise and fall.)