How did these writers know about it?
We have a death of a man who was martyred. oral traditions in illiterate societies.
We also know Mark was a compilation of pre existing written traditions. Paul tells us of other traditions and text in place in that time period.
We also know only a fraction of text and traditions survived to this day.
We also had a group of Hellenist who had been worshipping Judaism but would not fully convert, and did not want to be associated with these trouble makers.
So we have many different groups who all wanted Judaism light, these groups developed their theology around the martyred man and shared these traditions at Passover every year.
This is how and why the movement spread so far and so fast, and started equally in so many different geographic locations. Remember there was no center for where this started, its not like it originated in Jerusalem and spread from there. Its not what we see.
Someone probably told them right, they said that 'this group you see acting this way are the followers of Jesus who was crucified this long ago.'
It was not a long time ago. The stories were going around the Empire during his living memory of people there.
Paul was writing 15 years later and no one doubted his existence.
It probably not being unusual that revolutionary instigators were put to death, they'd have no reason to doubt a group's association with a leader.
Thing is, if one man started a riot trying to get people behind him to help overthrow the corrupt politicians and Roman oppressors, his actions would be perceived as a selfless sacrifice when he was crucified for standing up for the common man.