What do you call someone who thinks everyone is incompetent (doesn't know anything)?

"Having to explain every minute detail" is not explicit in the definition of the word, but it is the sort of behavior you might expect from someone who

patronizes: speak to or behave toward someone as if they are stupid or not important

other people.

That particular meaning works with the verb form (patronize) and adjective form (patronizing). Patronizer is out there, but it seems to get more use for denoting someone who engages in the more pleasant sort of patronizing (sponsoring, supporting, or bringing trade to). If you're willing to accept a phrase rather than a single word, you could describe someone as a "patronizing overexplainer."

If you're just looking for an adjective, another suitable word is

haughty: having or showing the insulting attitude of people who think that they are better, smarter, or more important than other people

though its definition also lacks the specific detail of feeling the need to explain minutiae to others.

P.S. I included links for the questioner because common search engines apparently proved too difficult. Positioning the mouse cursor over the red letters in italics and left-clicking with the mouse will open the pages with the quoted definitions.

I can't think of a good phrase for this, but if you are looking for descriptors, the two that come to mind are condescending and belittling.

Though these don't specifically incorporate this person's perception of the lack of knowledge of those he is belittling, at least they describe the resulting behavior.

Maybe you're looking for the Dunning-Kruger effect?

"The Dunning–Kruger effect is a cognitive bias wherein unskilled individuals suffer from illusory superiority, mistakenly assessing their ability to be much higher than is accurate. This bias is attributed to a metacognitive inability of the unskilled to recognize their ineptitude. Conversely, highly skilled individuals tend to underestimate their relative competence, erroneously assuming that tasks that are easy for them are also easy for others."

What about pontifical? Google defines it as characterized by a pompous and superior air of infallibility.