The question posed in the title of this thread is the type of illogical question a religious person would ask. I assume the term 'most people' used in the question refers to theists as opposed to non-theists, or to be more precise, atheists. Atheists don't think they are smarter than most people, but they know they are smarter than Religious people. The decline of religious belief across developed nations in the 20th century leading to more intelligent populations is indisputable !!! There is a good reason why the majority of the worlds academics don't believe in God, their is a correlation between increased intelligence and atheism.
Ironically, atheists possess far more humility and less pride and arrogance, as they have the ability to say 'I don't know' on a regular basis, opposed to religious zealots who attempt to answer the Universes most pressing mysteries with illogical and child like reasoning.