Burnt it? It was at a rolling boil most of the time and not a simmer. can you burn it?
Most assuredly you can burn anything! The idea is to reduce the Balsamic vinegar to syrup consistency, and the best way to do that successfully is low and slow. Bring the vinegar to a boil, reduce the heat to effect a gentle simmer, and let it go for quite some time. I usually set the timer for reasonable intervals (like 8 minutes) to remind me to check on the progress, and continue with other cooking chores. I also keep a wooden spoon in it (even though it's probably an old wives tale that a spoon in the syrup will keep it from boiling up and over). This works for all kinds of syrup reductions.
Just be sure you are not using the expensive, authentic Balsamico from Modena that costs a pretty penny. It's already naturally reduced.