This skit is trying to make the point that white people are privileged relative to African Americans in our society. It does so in a completely exaggerated and satirical way.
According to this skit, whites have an easier time of life in America both socially and economically. Economically, that is shown in the skit by "Mr. White" getting a free newspaper and a free loan. In social terms, it is shown by the party starting on the bus as soon as the black passenger gets off. This is a satirical way of trying to get us to see that white people in America have life easy compared to African Americans.
On the other hand, if you are so inclined, you could see this as a send-up of black fears of being oppressed. You could say that this shows Eddie Murphy's vision of how whites secretly have it easy and how white society conspires to oppress African Americans. You could see it as a criticism of the sorts of attitudes that allow African Americans to believe that, for example, AIDS was created by the US government to kill blacks.
The first time I saw the skit, I understood it in the first way that I mention. I still think that is the most likely meaning. But I can see where it might be understood in the second way.
It is probably best to start off with a few words on what Saturday Night Live is. It is a long running TV program that is based on various skits with a guest co-host. There is usually also a live performance from some artist. The format is comedic and this is the most important part. It can also be looked as a social commentary through the lens of humor.
In light of this the "white like me" skits explore the inequalities that exist between white and African Americans. Eddie Murphy played in this skit and he humorous goes undercover as a white person to see what these inequalities are. Keep in mind that this skit was done over twenty years ago where there was much more inequality. Words like, "the glass ceiling," which spoke of an invisible barrier that people of color could not pass in the corporate world were bantered around. In other words, racial issues were real.