The charges stemmed from Burr's murky involvement in a conspiracy to help himself to Spanish territory in North America at a time when that might have sparked a war between the United States and Spain. According to Doug Linder's account of the trial in the "Famous American Trials" series:
"Shortly after noon on May 22, 1807, the trial of Aaron Burr opened in Richmond. On the bench sat Chief Justice Marshall and Virginia District Judge Cyrus Griffin. Surrounding Burr was his team of defense lawyers including Edmund Randolph, John Wickham, Benjamin Botts, Charles Lee, and Luther Martin, a former Maryland delegate to the Constitutional Convention often called the "Thersites of the law." (In addition, Burr himself would play a major role in the trial, cross-examining most of the prosecution's witnesses himself.) The cast for the prosecution included George Hay, Caesar Rodney, William Wirt, and Alexander McCrae.
"While a grand jury awaited the arrival of General Wilkinson from New Orleans, Chief Justice Marshall considered both prosecution and defense motions.
"The prosecution, noting that 'the evidence is different now,' again moved for commitment of Burr on the charge of treason. The defense countered, arguing that to establish the crime of treason the prosecution must prove that an overt act of treason had been committed by the defendant in a war and that, under the Constitution, the overt act must be testified to by two witnesses and must have occurred in the district of the trial. When Marshall sided with the defense's narrow interpretation of treason, the prosecution knew it had its back to the wall.
The government -- inspired by a vindictive Thomas Jefferson in this case, foiled by a counterattacking John Marshall, according to some historians -- couldn't make the case stick. Burr was acquitted. The story of the trial is here.