Maintaining an appearance of affluence and wealth for the benefit of others.
In this world people don't respect you for accumulating wealth, they respect you for spending it. Most people who seem rich have very little wealth, they spend it all keeping up with the Jones's.
Jones is an extremely common surname in the United States and in this phrase is meant to be a generic term for the neighbors. The phrase makes much more sense when you say keeping up with the neighbors.
It is a common practice in suburbia for neighbors to be fiercely competitive, and to continually try to have the nicest of everything in the neighborhood.
Keeping up with the Joneses was the title of a comic strip that ran in many U.S. newspapers from 1914 to 1958 by Arthur R. (Pop) Morand. The strip chronicled his experiences living in suburbia.