I don't have an intuitive understanding of why I'm coming across as arrogant.

Think in monkey-terms. Humans are just hairless bipedal apes and status matters, a lot.

Statements of what you perceive as (fairly obvious) facts have implications, in particular social/status implications. Human conversations are simultaneously an exchange of information and an exchange of signals. Most people automatically process these signals on the slightly subconscious level and respond with signals of their own without necessarily being aware of it. Women, in particular, are quite adept at this.

People in whom the signal-processing mechanism is inefficient, miscalibrated, or just plain broken have trouble with navigating social interactions. The interaction flows on (at least) two levels but the invisible layer is malfunctioning and if you don't even know it exists you are confused why the overt information-exchange layer is doing so badly.

I suspect that if the subconscious mechanisms are not doing their job, you have to bring the signal-exchange layer into the territory of the conscious and explicitly manage it.

Accept that every conversation has two layers even if you don't see one of them. Evaluate all statements (verbal + body language, etc.) on two levels: (1) what does it say; (2) what kind of signal it sends, what does it imply.

To return to your original question, on the overt information-exchange layer you see your statement "I am smarter than almost everyone here" as a neutral fact about the world which you believe is true. Now, analyze that statement on the signal-exchange level. What does it imply to hairless bipedal apes?