How to fix a MacBook Pro display that turns off?
This is a pretty vague question, but try some of these things: 


Does the monitor work properly on other computers? (Probably you should check this first.)
You don't say if the external monitor is an Apple monitor or not, but are there drivers for the external monitor? If so, are they current? 
Did you change anything in your system prior to the external screens breaking? 
What happens prior to the external monitor turning off? Are you getting an error message? How long does the external screen work before it turns off? Is it an LCD or an older CRT monitor? 
I was thinking this could be an issue with the monitor overheating and switching off, but it's currently not the summer where you are. Does the monitor have a fan that stopped working?  
Does this happen when you boot into another user? What about if you boot onto another volume?