This is a take on a quote from Samuel Taylor Coleridge, which talks about the "willing suspension of disbelief" when immersed in a work of art. By art is meant poetry and literature in particular. Coleridge's statement is often quoted in application to these, and I have often seen it quoted in application to movies and theater, too.
The idea of the original quote is that the reader or viewer should allow him or herself to be so immersed in the world of the poem, novel, or movie that there a complete belief in that world. Have you ever been to a movie with someone who insists on finding the flaws in the movie or who keeps explaining how the special effects were done? That person has no suspension of disbelief. He or she is trying to deconstruct the movie, rather than to be immersed in the world of the story.
Now, as for what Lionel Trilling has to say, I believe his idea is that people are acting so outrageously in today's world that in order for us to even process their behavior, we must suspend our disbelief in their behavior. Lionel Trilling is, of course, a humorist, and my guess is that this quote is not meant exclusively as a social commentary, but also as humor.
I do not know anything about the context of his statement, but as I read the news every day, I can think of any number of situations for which his comment is appropriate. When people pretend their child has disappeared in a balloon, when people crash White House dinners, when people go on television shows to talk about their messy personal lives, all we can do is sit there and take it in without trying to figure it out because there is no figuring out this kind of behavior.