Why were Sacco and Vanzetti innocent and what was the significance of the case?

Nicola Sacco and Bartolomeo Vanzetti were two Italian American anarchists convicted of murder and armed robbery in 1920, crimes for which they were executed seven years later. Their executions came at a time in American history when there was widespread fear and mistrust of radical politics, which was associated in the popular mind with violence and disorder. The case was a polarizing one, with many high-profile public figures campaigning against what they saw as a major miscarriage of justice.
Even at the time of their conviction, the case against Sacco and Vanzetti was seriously flawed in many important respects. Eyewitnesses for the prosecution gave conflicting evidence of the defendants' involvement in the crime and in some cases even changed their story. The prosecution case against Vanzetti was especially flimsy, as many eyewitnesses testified that he was at work during the time of the fatal robbery.
The case has been pored over by countless lawyers, jurists, and scholars ever since. Most seem to agree that there was reasonable doubt as to the defendants' guilt. Others have suggested that Sacco did indeed pull the trigger but that Vanzetti was not involved in the actual robbery. In any case, the general consensus is that Sacco and Vanzetti did not receive a fair trial, not least because of the open bias and partiality of the presiding judge, Webster Thayer. The lack of basic fairness in the conduct of the trial was acknowledged in 1977, when the then governor of Massachusetts, Michael Dukakis—the future Democratic presidential candidate—formally proclaimed that Sacco and Vanzetti had been unfairly tried and convicted.
What is significant about the case is that the defendants were on trial for their deeply unpopular, "un-American" political opinions as much as anything. The prosecution persistently used Sacco and Vanzetti's avowed radicalism as a means of establishing circumstantial guilt. Their background as Italian immigrants, their broken English, and their social views were all ruthlessly exploited by the prosecution to convey the impression that, even if they were not guilty of the actual crimes for which they had been charged, they were still dangerous radicals—dangerous foreign radicals—and needed to be punished accordingly.
https://www.theatlantic.com/magazine/archive/1927/03/the-case-of-sacco-and-vanzetti/306625/