It simply means being proud of the city or town one lives in. 

It's sometimes used ironically, because in the United States it is normal for people in small towns to show civic pride even when their town is not outstanding or very different from the towns around it. 

A typical example of "civic pride" would be the signs near Lancaster, Wisconsin that say "LANCASTER: City of the Dome." This is because Lancaster--population 3,686--happens to be the county seat and is the location of the county courthouse. The courthouse has a small cast-iron dome on it. The courthouse building is the only monumental structure in Lancaster, so people are proud of it. There are probably a thousand courthouses like it throughout the United States.

People who show civic pride do things like organizing and taking part in the Fourth of July parade, participating in "service organizations" like Rotary and Kiwanis, joining the Women's Community Committee and beautifying the town park by planting flowers in it.