Clash's multiplayer mode lets you attack and visit villages that belong to other real life Clashers. By attacking, you can raid their resources like gold, elixir and, at higher levels, dark elixir. This lets you build and upgrade your village much faster than by simply saving up resources from mines or collectors. It's also a lot more fun!
By winning battles, you stand to gain trophies. The more stars you get, the more trophies you win. If you lose the battle, you also stand to lose trophies. Once you have enough trophies, you can gain entry into the Trophy Leagues, which let you win even more precious resources.