What happened to the outside world in Danganronpa? In the last episode on Danganronpa, we get a glimpse of the outside world where a giant bear in wrecking havoc in the city. People are wearing bear masks and rioting in the streets.

What exactly  happened to world? Was the despair that Monokuma talking about actually a disease? Was everyone related to the characters in the school already dead?

PS: I'm not familiar with the game or novel I have only watched the anime.
I have played the game, and yet I can tell you the answer isn't the most clear there either. All I got from there is that the incident started on Hope's Peak Academy and propagated an extreme state of unrest to the outside world. However, I've heard things are explain further in the sequel, which I haven't played nor followed a Let's Play of just yet. Taking a look at the Danganronpa wiki, they explain it quite well there:


  "The biggest and most despair inducing incident in all of Hope's Peak Academy ’s history" was only the beginning to what would be the world's end. The biggest and most despair
  inducing incident in all of Hope's Peak Academy’s history caused the reserve course students to rebel…
  
  At first, it was optimistically thought that everything would calm down soon, but the situation worsened and quickened its pace. The movement happened on the Internet, and formed a community of its own. 
  
  Soon, not just students were involved, but came to include people of various races and nationalities in its development and spread off the internet and into real life. 
  
  At the beginning of the movement, it involved demonstrations regarding social issues, but as it spread through the world, its abnormality began to show. At some point in time, the motives and modus operandi changed, and left behind only meaningless destruction and violence.
  
  Strong people killed weak people…
  
  Weak people murdered even weaker people…
  
  Weak people formed factions and lynched strong people
  
  As the violence and death spread, people became desensitized. As if time had gone backwards, people took death to become a certainty. The media reports overflowed with death, and people ate as they watched. By the time the world had noticed the abnormality, it was already too late.
  
  The overpowering “despair” became a huge wave that swallowed up the world in the blink of one’s eye…
  
  Soon, terrorists and coup d’etats sprang up and caused the despair that is war. It wasn’t war that happened due to the clash of ideals, religion or profits…It was just war. 
  
  Pure war. 
  
  Therefore, no way to solve the problem was found. However…how did what was originally a students’ movement develop into such a despair-worthy situation? This was because of…the existence of “a certain group of people”.
  
  It revolved around the “certain student” who led Hope's Peak Academy to its destruction; a group called “Super High School Level Despair”.
  
  Their talents, recognized by Hope's Peak Academy, were used not “for the sake of causing hope for humanity”…They used their talents “for the sake of causing despair for humanity”. Those who held great power brainwashed ordinary citizens to spread despair… 
  
  Those who had talent with computers created software to spread despair…
  
  Those who had great influence over people created new ideals to spread despair.
  
  This is how “the biggest and most despair inducing incident in all human history” was produced. Indeed, as long as “Super High School Level Despair” exists, this despair will not end…
  
  “The biggest and most despair inducing incident in all human history” will not end…


And that's apparently the in-game description, though being as I don't remember reading through that, it's probably from Super Danganronpa 2.

To read more just check the source where I took that from:
http://danganronpa.wikia.com/wiki/The_World%27s_Most_Despair-Inducing_Incident