If the verb to "construct" means to build something, then to reconstruct means to build it again. You can reconstruct a building, a city, or even an idea.
If a house is destroyed in a fire, earthquake, or flood, you might try to reconstruct it — or you could build a totally new one. Either way, though, you could still be said to be reconstructing your home. Some buildings were destroyed so long ago that we don't know what they really looked like. The same can be said for whole cities and cultures. Still, archeologists try to reconstruct how those cities and cultures appeared at the time that they were flourishing. Linguists reconstruct long-dead languages. Detectives try to reconstruct crimes in order to figure out who committed them.