I'm not so sure that people in the past would have been "fine" with public executions per se, but rather you seem to be asking why was it an accepted practice that executions were performed in public.

The answer to that is going to differ based on the time period and geography, but at least for medieval and early modern Europe a public execution served a judicial purpose: it sent a message that the sovereign and/or state could and would exercise lethal power in the event that laws were broken. Executions were staged as a theatrical event intended to induce fear of the state in those who saw it.

The book most often cited as a theoretical basis for this claim is Michel Foucault's Discipline and Punish: The Birth of the Prison (1975). The study is an explanation of how the modern prison system emerged, but Foucault points out that early modern forms of justice were publicly administered for "juridico-political" reasons:

A successful public execution justified justice, in that it published the truth of the crime in the very body of the man to be executed...The public execution is to be understood not only as a judicial, but also as a political ritual. It belongs, even in minor cases, to the ceremonies by which power is manifested (44, 47).

This extended beyond capital punishment; the purpose of many judicial sentences was precisely to send a message to others about justice and the law. Being sentenced to the stocks served a similar function - public humiliation intended not so much for the accused as for those watching. Foucault's argument is a bit more complex and has additional facets (e.g. crimes against the law were seen as attacks on the sovereign body, and thus an execution reconstituted that sovereignty), but that's his argument in a nutshell.

The rest of the book addressed precisely why Foucault thinks that process changed. According to him, public executions had unintended consequences. They sometimes garnered sympathy for the accused and fueled resentment toward the state, as excessively violent punishment might be viewed as an arbitrary exercise of power. It's notable that the context Foucault draws on is that of early modern France, where breaking on the wheel was a sanctioned form of execution. It was, as Foucault puts it, "inefficient."

At some point in the late eighteenth/early nineteenth century, the focus shifted from controlling public sentiment to controlling individual bodies. In other words, states stopped controlling crime by "setting an example" and started controlling it by locking people up in prisons so they could be monitored around the clock. This was considered a more direct, and thus more efficient, method of control.

One of Foucault's most well-known concepts is panopticism, exemplified by Jeremy Bentham's panopticon - a penitentiary design intended to ensure that inmates are being observed at all times:

Hence the major effect of the Panopticon: to induce in the inmate a state of conscious and permanent visibility that assures the automatic functioning of power (201).

If prisoners could be watched constantly - or even more efficiently, if they could be made to think they were being watched constantly - they would be less likely to engage in "criminal behavior." The intention of the prison was to reform the individual, not send a message to the collective.

That's the big message of Foucault's book: early modern systems of justice may have been physically brutal, but modern forms are no less insidious in their attempts to institute control. Though it should also be pointed out that this process was neither clean nor immediate; France, for example, held its last public execution in 1939, and the last one in the U.S. was in 1936.