This is a social psychology question and the answer doesn't have as much to do with the particulars of what a programmer does as the social environment "programming" lives in.

The easiest concept regarding this phenomenon is self selection bias. Certain groups of people are drawn to, and others are excluded from certain professions. These groups are usually defined by demographics and personality traits. This results in a self-preserving system with its own gatekeeping, leading to a self-preserving subculture.

Obviously this isn't unique to programming. Every human group regardless of how that group is defined have in-group and out-group biases which perpetuates us and them identities in our minds. Everyone has these to varying degrees.

If we want to talk about why programming seems to select for the trait of arrogance, we have to speculate.

I think it could be related to the esoteric nature of professions like electronics engineering and programming. Things these professions work with do not have moving parts. Their internal workings can't be guessed by their physical appearance, nor from their immediate function. This might be creating a feeling of magic, as in any advanced enough technology blah blah... You know what I mean.

Perhaps programmers start to believe in the magic themselves, or to take it seriously when they are called "tech wizards" etc.

In return, non-programmers are probably happy to benefit from the "magic" without going into the nitty gritty of all the frustrating grunt work required to make things work, and they exclude themselves from the profession.