"to betray" is a flowery, or fancy, way of saying "indicates" or "shows evidence for". It's not used in its literal sense here because a remark is not a person so it can't literally betray. :) To be "staggered" literally means that you've been hit (by something) so hard that you're stunned and you're unsteady on your feet (you've been made to stagger). Information can be so surprising that it figuratively "staggers" the listener. In this case the information is the degree to which the person seems to have contempt. It's not a moderate amount of contempt, it's a staggering amount. Contempt means having a very low opinion of. So the sentence means, in other words, "His remarks show evidence that he has (or indicate that he has) a level of contempt for the truth that utterly shocks me." Or "He is a big, fat, liar."

"Betraying someone" actually means exposing or sharing hidden information about that person. "Your words betray you" or "your remarks betray you" means that your words indicate something that you want to keep hidden. For example, if I am secretly in love with someone, but I talk about him all the time, then my words betray me - they indicate my real feelings. "have contempt for ________" means to feel that ______ is worthless or stupid. "To have contempt for the truth" means that you don't think truthfulness and honesty are important and that there is nothing wrong with lying. In the US, most people think that lying is bad. If you don't think lying is bad, you may want to keep that hidden. "His words betray a staggering contempt for the truth" means that "he thinks lying is fine. He doesn't want people to know that he thinks lying is fine. His words make it clear that he really thinks that lying is fine." ("staggering" is an intensifier.) Or, in the end, as Steve said "He's a big, fat liar."