The best definition I have heard of cultural hegemony is that it is the art of getting people to "participate in their own oppression." Marxists argue that the ruling class achieves cultural hegemony by getting the other classes to do this.
According to Marxist thinking, the dominant class puts forward a set of ideals that it portrays as beneficial to all people. An example of this would be the American ideal of individual responsibility and hard work. The classes that are not dominant come to believe in this idea. We can see this quite clearly in the US -- even people of the working class believe that Americans can get ahead if they try hard enough.
Marxists say that these ideals actually only help the dominant class. For example, they would argue that the ideal of personal responsibility helps the dominant class by making the other classes think that they really can get ahead with hard work. It makes the other classes look at poor people and blame them for their own poverty rather than blaming the dominant class that keeps them in poverty.
In this way, Marxists say, the dominant class gets the others to participate in their own oppression -- it gets them to believe that poverty is the fault of the poor and encourages them to accept the system that exploits them.