I know very little about Republican Spanish governance except that they may have killed a few catholic priests. What exactly did they do and why do socialists always use it as their one example? Is this a result of its relatively short and uninteresting existence or is there actually some merit to this claim?
So, the way you've posed this question is difficult to answer from a purely historical perspective, because you're asking less about the Spanish Republic itself and more about how it is remembered and how that memory gets used. The Spanish Republic is a bit like a political Roschach test - what you see when you look at it is going to vary a great deal depending on your own beliefs. Analysing this isn't exactly divorced from history, but makes any attempt inherently subjective. So I'm going to tread a fine line here - I'm essentially going to tell you what I think, with the caveat that I'm happy to do my best to provide a more straightforwardly analytical follow-up about aspects of the Spanish Republic as it actually existed if you have specific questions.
To begin with, 'short and uninteresting existence' is only half right. The Spanish Republic lasted less than a decade (1931-39), but it was certainly an exciting ride. Not only was the Republic itself conceived of as a bold, progressive democratic project, it was launched at a time that democracy was crumbling across Europe. This made Spain's experiment more prominent than it might have been on a world stage, appearing as an example of hope amidst the rise of authoritarianism elsewhere. Spanish democracy, however, had fundamental issues from the start. Spain was deeply divided culturally and politically, and the way the electoral system had been designed meant that narrow victories translated into a great deal of political power. Moreover, much of Spain's political left and right did not ever really acknowledge the legitimacy of the other side. For the left, the Republic was an inherently progressive project, and any deviation from this - such as a winding back of many early reforms after a conservative electoral victory in 1933 - was not just politics as usual after a change of government, but an attack on the very purpose of the Republic itself. Equally, much of the Spanish right viewed the Republic itself as illegitimate, and saw many of its reforms as striking at the heart of Spanish culture, notably the position of the Catholic Church in society. This meant that neither side was particularly willing to sit quietly while the other was in power. Political conflict was increasingly not contained within the sphere of formal politics, with political organisations acting outside of parliament and government, sometimes subverting the state itself in order to do so. Most famously, militant strikes and uprisings in the mining regions of Asturias in 1934, aimed at toppling the right-wing government, were met with extreme force from troops shifted in from North Africa.
So Republican Spain was hardly dull, even before 1936, when an election victory for the left-wing Popular Front eventually led to an attempted military coup and ensuing civil war that lasted until 1939. This was a violent, ugly conflict, as civil wars tend to be. Neither side had clean hands at the end. You mention the deaths of Catholic priests, which understates things. Spain saw some of the most intense anti-clerical violence in modern history, with thousands of deaths - in some regions such as Catalonia, around a third of the pre-war clergy were killed. Both sides also executed or imprisoned large numbers of perceived political enemies within their territory, particularly in the early months of the conflict. The 'Nationalist' side, led by Franco, were unquestionably worse in this regard, but the Republicans carried out significant atrocities. There is some debate about the extent that these atrocities should be regarded as 'Republican' - particularly in the early days of the war, the central government had little authority in the Republican zone, and much violence was supposedly carried out by anarchist or other groups of their own accord. This picture is debatable, but its certainly the case that the bulk of violence in Republican Spain took place in the first months of the war, while Nationalist atrocities continued throughout the conflict (and after). This does however mean that the historical reputation of the Republic is somewhat better than it might be.
The Republic's reputation is also enhanced by the international context of the conflict. It was easy to read European politics into the situation - here was a democratic government under attack from a right-wing, (quasi-)fascist uprising, directly supported by Hitler and Mussolini. Crucially, the resistance to the coup attempt and then against international intervention seemed to be the first moment anywhere that it seemed possible fascism could be defeated - while democracy had died with a whimper throughout Europe, Spain seemed to prove that anti-fascist unity could actually work. As such, the Spanish Civil War became a major cause for the anti-fascist left across the world, the flashpoint of a much wider ideological struggle, similarly to how the Vietnam War became emblematic of Cold War conflicts. Solidarity movements abroad grew to a scale never really seen before, raising huge amounts of money to provide support and humanitarian relief for the Republic. Even more importantly for the question of memory, huge numbers of foreigners went to Spain to fight for the Republic, approximately 35,000 all told. This remains an unprecedented mobilisation of foreign volunteers for a single conflict (and, for what it's worth, the subject of my research!), and these volunteers represented a crucial tie between their home societies and the Republic. Their sacrifice - about a quarter of them were killed - in the name of their beliefs made for strong and lasting connections to the cause they fought for. These volunteers are still commemorated by many groups and communities (particularly on the left), who naturally tend to have very supportive views for the Republic itself.
The varied political make-up of the Republic itself also lends itself to being celebrated by a broad spectrum of supporters. The Popular Front government was certainly progressive, but was not unambiguously 'socialist', and many of its reforms were well within the bounds of what would have been considered 'centre-left' at the time, such as improving working conditions and developing a secular education system. Yet the civil war provided an opportunity for far-left groups to put their ideas into practice, particularly in the early months of the war which saw effective control pass from the central government to local organisations, most famously in regions like Catalonia where the strength of anarchist groups led to the collectivisation of some industrial and agricultural production. Other groups felt themselves to be validated in other ways - the Communists' advocacy of disciplined anti-fascist (and anti-Trotskyist) unity, which saw their influence increase dramatically over the course of the conflict, could be taken as proof of the efficacy of their methods and leadership. While unity did broadly prevail during the war, defeat meant that recrimations followed, with each faction claiming that the decision not to follow their preferred strategy led to the Republic's defeat. This is where Orwell's position in Homage to Catalonia fits in - that in abandoning the revolutionary aspects of the struggle as supposedly advocated by the Communists, the war could not be won. These debates meant that just about any progressive political group could point to Spain as proof or validation of their ideas.
Lastly, for socialists and communists who lived through the mid-twentieth century, the memory of Spain offered a seeming sanctuary from the tarnishing of their ideals. With revelations about Stalin in the 1950s, followed by the actions of the Soviet Union in Hungary and Czechoslovakia in 1956 and 1968, it was difficult to maintain the same kind of idealism that had driven many of them to embrace their political beliefs in the first place. Spain was different - it was a straightforward fight against fascism in defence of democracy (and/or socialism). To my mind, it's useful to think of Spain as a martyr in the longer fight against fascism - because the Spanish Republicans were defeated before they could achieve their goals, it was the tragic victim of fascist expansionism and Western appeasement. Spain became known as the 'Last Great Cause', a fight which required no awkward moral compromises or Cold War realpolitik. Obviously, this did not reflect the reality of the Republic's existence, but was nonetheless a powerful mythical device that continues to resonate today.