Basically, the duel between Burr and Hamilton resulted from decades of provocation and slander from Hamilton, including spreading rumors that Burr was sleeping with his own daughter. Hamilton shot at Burr during the duel, but lied about it afterwards and then died like a coward. https://twitter.com/California_Tory/status/1275615148787372032
The press, very friendly with Hamilton (who often leaked things to them), did their best to smear Burr after the duel. Hamilton's rumor campaign had already destroyed Burr's relationship with Jefferson, so Burr resigned after leaving the VP office much better than he found it.
The accusations of "treason" against Burr after the duel were totally false. After resigning as VP, Burr organized a few dozen soldiers and engineers for a filibustering (land/asset seizure) expedition to Florida, which at the time was a normal way to grow the country.
When the US purchased Florida, Burr traveled to New Orleans to make plans for a filibustering expedition in Mexico. While he was there, a disgraced general looking to get back into favor with Jefferson started spreading rumors about Burr's supposed plan for a coup.
To support his accusations, the retired general forged a letter claimed to be from Burr, which detailed plans to create an independent monarchy in the South. The press went haywire with these rumors, and Jefferson put out a warrant for Burr's arrest.
Burr turned himself in several times, but each time the federal judges released him because no crime had been committed and there was no evidence to support the accusations. One judge refused to hold Burr because he was concerned that the feds were trying to have him killed.
The general's story became more extravagant, claiming that Burr was backed by Britain and France and that he wanted to burn New Orleans to the ground. He used these claims to declare himself military governor of the city and arrest his critics.
Eventually federal agents caught up with Burr, enlisting the local militia to help arrest him. After Burr was arrested, the militia refused to leave his side because they were concerned that the feds were going to murder Burr (who was extremely popular) in transit.
One of the militiamen was reported as saying that they all knew that the accusations of treason weren't true because if Burr was actually raising an army to torch New Orleans, they would have joined it.
Burr was taken back to Richmond, where Chief Justice of the Supreme Court Marshall insisted on presiding over the trial himself, because he was concerned that Burr would be murdered by Jefferson or Hamilton's men if the trial was held in DC.
The trial was a joke. The government's only witness, the disgraced general, was drunk on the stand and revealed that the only piece of evidence, the letter, was a forgery. Future President Andrew Jackson testified in Burr's defense. Burr was acquitted of all the charges.
Aaron Burr was a great man who loved America. He was the victims of one of the earliest and longest-running smear campaigns in American history, a smear campaign that still continues to this day.