In fact, Aaron Burr was tried twice for the same "crime" of treason and acquitted both times. The short answer is, in the eyes of the Law he did not foment a rebellion; and today nobody really knows for sure what Burr's actual intentions were.
Rather than trying to pry New Orleans and parts of Louisiana away from the US, Burr could have been intending to 'filibuster' in Mexico (Texas) and carve out a new state. If this is what he was trying to do, he was actually ahead of his time. Sam Houston, with the explicit backing of President Jefferson, did accomplish the same feat in the 1830's.
Burr's main co-conspirator was General James Wilkinson, who was the military commander of New Orleans. It has been proven that Wilkinson was in fact taking bribes from the government of Spain. It is a strange fact that even though Jefferson and Madison suspected this, Wilkinson was allowed to be one of the chief US generals during the War of 1812. It seems that Wilkinson panicked and 'ratted' on Burr to Jefferson, saving his own hide by doing so. However the word of Wilkinson in terms of what Burr was actually up to is not very credible (or at least it wasn't to Chief Justice John Marshall and the jury that presided over Burr's Washington D.C. trial).
There was a militia base set up on Blennerhasset Island on the Ohio, where it appeared that there was a military operation being set up. This operation was busted up by the Ohio militia and the owner of the island was tried in D.C. along with Burr. So there likely was at least a military operation being set up. Theoretically, Wilkinson would have jumped in with his US troops in New Orleans using some pretext. The question is, would Burr and Wilkinson have moved against Mexico, or would they have moved against the US and presented Louisiana to Spain?
Burr - who was an incredibly charismatic figure and was greatly admired especially in the West even though (or because) he killed Hamilton - nearly dragged down several leading American political figures along with himself. Henry Clay was the lawyer to got Burr acquitted during his first trial. Later Clay became convinced that Burr must have been guilty of something and he rued having represented Burr. Also Burr spent much time with Jackson in Tennessee and Jackson always was supportive of Burr. Some people have speculated whether or not Jackson would have joined Burr's expedition if it had actually been activated.