If you are sitting around quietly with a group of people. If one person says something interesting. Do you repeat it to the group, or assume that the rest of the group heard it because they were all sitting in equal proximity to the person that made the original statement? Split horizon would be the assumption that everyone else heard the statement and you have no need to repeat it.
From a network perspective it is very similar. If a router hears of a route, it may repeat those routes out another interface, but not the interface it heard the routes from. This is because if there are multiple routers, directly connected out that single interface, the assumption can be made that they all have equal access to one another. At least that is the assumption.
There are situations that it is not true and split horizon should be disabled. If a route is received on an interface, the router "typically" should not send that route out the same interface. The route should be sent out the other interfaces.
I think its fair to say that Paul and Armen have descibed Split Horizon in a clear and concise way.
Just to add to what they have said above, you may find this Wiki link worthwhile:
Wikipedia ... the greatest thing since sliced bread... and TCP/IP
All the best,