Why does Mana-Burn exist in Magic: The Gathering?
Mana burn has been removed from the game in the magic 2010 rules update. From the comprehensive rules glossary:

Mana Burn (Obsolete)
Older versions of the rules stated that unspent mana caused a player to lose life; this was called &quot;mana burn.&quot; That rule no longer exists.

When the rule was still in effect, you lost all unspent mana in your mana pool at the end of each step and phase (not turn), and you lost life equal to the amount of mana lost.
As for the reason why it was there: As you said, it was meant to punish anyone who didn't manage his or her mana well. Also, there were a few cards that had the mana Burn rule in mind. They would add Mana to a player's pool during on of his/her game phases. Some would reward you for being tapped out or punish the opponent for not being tapped out, mostly in the Prophecy edition.
Why was it removed? As you noted as well, it was practically irrelevant a rule as it only very rarely happened, and when it happened, it confused most players about the details of how it worked exactly.