I cannot comment on Germany or Italy, but I can shed some light on Spain's situation. For the best readily accessible source for the ideological background for the Spanish Civil War I would recommend Paul Preston's books. His Spanish Civil War would suffice in a pinch, but The Spanish Holocaust is stellar regarding how Spain split into ideological camps in the years before the war began. He describes a process of "polarization and radicalization" that eroded the ability of moderates to stay in control while encouraging radicalism on both ends of the political spectrum.

Part of that split was a series of accusations and recriminations by those on the right. Freemasonry was one of a list of targets that also included Jews, communists, anarchists, regionalists, antimilitarists, democrats, republicans, unions, and much more. I would assert that the rejection of Freemasonry was rooted in the regime's desire to court Catholic support. Freemasonry and Catholicism have had a long history of conflict. The more recent pronouncements on the issue prior to the Spanish Civil War would be Leo XIII's Ab apostolici in 1890 (though that encyclical targeted Italy in particular) and the 1917 Code of Canon Law that made membership in Freemasonry an automatic excommunication. Some in Spain picked up that torch and took it much further.

The most salient example is Fr. Juan Tusquets, who was a successful author prior to the war breaking out. His book Orígenes de la revolución española was a translation and extension of The Protocols of the Elders of Zion, an anti-Semetic hoax created to justify attacks on Jews. The original document contains anti-Masonic portions, and Tusquets (and others) expanded and adapted the Protocols for a Spanish audience. He emphasized the Protocols allegations of Jews and Masons colluding and combined it with an anti-communist message. To "verify" his claims he conducted raids on Masonic buildings and individuals and compiled a dossier to "prove" his case. The result was that Tusquets asserted in newspaper articles and books that there was a "Judeo-Masonic-Bolshevik" conspiracy to destroy civilization--and Spain in particular (there was also a “Judeo-Masonic-Mahommedan-Bolshevik” variant of the conspiracy. Given the incredibly tiny Jewish and Muslim populations in Spain as well as the fictional nature of the conspiracy it is remarkable that such ideas caught on).

Sadly, Tusquets was not alone. His was just one book that contributed to the demonization and dehumanization of parts of the Spanish left. General Emelio Mola wrote a series of memoirs while under house arrest for using excessive force against a student demonstration, and was clearly influenced by Tusquets. The most virulent of his works was El Pasado, Azaña y el porvenir. There were also translations of the anti-communist journal out of Geneva, the Belletin de l'Entente Internationale contre la Troième Internationale--which was related to Goebbel's Ministry of Information. Yeah, that Goebbel's. Another was from Léon Poncins: Las fuerzas secretas de la Revolución. Again Freemasons and Jews were castigated. There were more as well. As Paul Preston states:

The ultra right-wing press in general regarded The Protocols as a serious sociological study.

That these works were taken seriously had fatal repercussions. Franco's regime wanted to curry favor with the Church, and did so by a number of means. One of them was executing large numbers of people they saw as opposed to the regime and the Church. Masonic membership was one of a long list of excuses that could result in your execution during (and after) the Spanish Civil War. It is also tragic that many clergy participated in denouncing people that were then imprisoned or executed executed as well as providing ideological justifications for the conflict before the war began. This isn’t to say that there weren’t a number of good priests and bishops that tried to mitigate the violence, but that any clergy would twist Catholicism to justify murder is abhorrent. Sanchez states that

[the] warping of Christ's message is what makes the clergy's support of the Nationalists and their silence in the face of the reprisals so reprehensible, and it makes the anticlerical fury seem justifiable (although in fact the fury in most cases preceded the support; yet the anticlericals were protesting years of Christian neglect)." While there were countless good and merciful priests who tried to live the Christian ideals of love and brotherhood, it was the ecclesiastical hierarchy that attracted attention by their scandal of silence, and good men everywhere suffered because of it. Therein lies one of the great tragedies of war. (*The Spanish Civil War as a Religious Tragedy, pg 116)

Regarding the executions he also argues that

The anticlerical fury [the murder of 6,832 clergy in Republican territory during the war] was a visible indictment of Catholic attempts to channel the essence of Christianity into narrow parochial ends. And worse, those Catholics who were not sacrificed to the fury condoned by their silence unchristian, inhuman reprisals against victims of circumstance, and they publicly lauded and supported a regime built in large part on oppression and special privilege. They became the clergy and laity of the church of vengeance, and they lost the opportunity to form the truly Christian church of reconciliation (Pg 199)

Overall the demonization of Masons in Spain was part of an overarching effort to dehumanize opponents of the rightist groups that eventually made up the Nationalists. It should be noted that such rhetoric went far beyond Vatican statements on the subject. To be fair, there were also leftist elements that applied similar rhetoric to dehumanize and vilify rightists, and both sides of the Spanish Civil War featured a great deal of violence that targeted noncombatants (again, Preston's Holocaust is excellent for a narrative of this subject). Still, Fr. Tusquets and others clearly contributed to a climate that justified the execution of Masons and others.

A curious endnote to Tusquets' life is that he was indeed very popular in Spain before the war broke out and collaborated with the Nationalist uprising, but when he was later offered influential positions in Franco's regime he declined them. He went into teaching and ceased publishing. One would hope that he saw the carnage of the war and recoiled from the rampant violence that was partly inspired by his words. There are other explanations for his refusals to join Franco's regime, but I am hopeful that he saw just how damaging his words were and wanted to go no further with them.

As always, followup questions from OP and others are encouraged.