Code sharing allows an airline which would not normally have traffic rights on a certain route to offer a more complete service, for example through fares and FF membership benefits from origin to destination, terminal sharing, through baggage check, and so on. It can also offer economies of scale.
There's lot more to it, that's in a nutshell. There are lots of other types of codeshare, leased and blocked space, capped and uncapped, it also goes into the realms of franchise operations, wet/dry lease. Codeshare is a very general term covering a multitude of different types of arrangement.