Why is it important to round answers to significant digits?

Rounding stops silliness from just looking at a calculator display and telling the person something misleading. You would not determine the number of people on a school campus and say “There are 3567.45 students.” However, if you were doing some estimates and the display on the calculator suggested that number then you would round to what you are certain is correct plus one more place for the estimated digit. This is determined by using the number of significant figures of your “weakest” measurement (the measurement with the least number of significant figures).

What does that have to do with uncertainty?

Measurements are only as accurate as the tool allows. All measurements have uncertainty; that is, at some point the tool requires the user to make a guess (estimate) as to the last digit. Choose the wrong tool and you could end up causing trouble, for example, if a person wants to build a patio 4 meters by 4 meters then they would not use a tool that measures length to the nanometer. At the same time, the person would not use a tool meant for measuring kilometers.

Most important, if you are measuring something then you want to be certain that you are close to the desired measurement. For example, if you are measuring your height then you would not say to someone “I’m 0.002 kilometers tall” nor would you say “I’m 200 centimeters tall” because the last one could be done with more accuracy with a different tool, i.e. 172.72 centimeters.

Further Reading
http://www.thechembook.com/wiki/index.php/Significant_figures