From a practical perspective, they help to keep the rain water that drips off of a building's roof away from its walls. Water on the walls will become trapped water inside the walls, which quickly leads to mildew, mold, and rot. Similarly, eaves keep icicles from forming directly on a building's outer walls (an icicle on the wall can become an icicle dripping down the wall, leading again to water inside the wall).

Eaves that stick out 6" are just barely deep enough to keep rain off; a 9"-12" overhang is better. The Carriage House eaves shown here are 18" deep. Copper gutters - now worn out and removed - originally sat in the curved brackets running along the edge, adding 4" more depth as well as redirecting the water.
Eaves are also for play, of course - they make the carriage house fun to look at. And without eaves, this building would just be an awkward box with bumps. The length of the eaves, their edge moldings, and the rows of brackets underneath all come together to create a roof that visually shelters what's inside and delights the eye. The corbels facing both ways at the ends of the dormer windows and the at the barn's cornerssee the first photo) are just frosting on the cake.

That's a lot of creative 'green' stuff for an ordinary building detail to do.
Here is the whole series:
Part 1 - http://www.jgrarchitect.com/2008/04/building-to-weather.html
Part 2 - How does the carriage house work with the sun to minimize wind chill?
http://www.jgrarchitect.com/2008/04/building-to-weather-2.html
Part 3 - Why bother with a cupola?
http://www.jgrarchitect.com/2008/05/thats-pretty-amazing-cupola-with-all.html
Part 4 - Eaves? they're important?
http://www.jgrarchitect.com/2008/05/eaves-for-work-and-play.html
Part 5 - How a floor plan makes a difference:
http://www.jgrarchitect.com/2008/06/no-tech-warmth-with-low-tech-aid.html
Part 6 - A look at how these concepts were used at the Big House:
http://www.jgrarchitect.com/2008/09/big-house-is-green-too.html
Part 7 - Shutters:
http://www.jgrarchitect.com/2008/03/shutters-ok-i-know-that-they-were-for.ht
Read Building to the Weather - Part 3.
Read