He was not used to public speaking; make a complete ass of oneself

[Tdol]

I don't also understand the "public speaking" part of the example

If you have to speak in public, for instance at a wedding, big dinner, or meeting, that is public speaking. If you have to stand up to give your speech, you could call it public speaking.

 

[Barb_D]

Oh my.

You can't really become "used to" something without doing it many times or having long exposure to it. However, doing it many times doesn't mean you are comfortable with it (used to it).

I used to live in Maine so I am used to cold weather.
I used to visit New Orleans every summer but never got used to how hot and humid it was.

I used to go New York City a few times a year, but I never became used to the subway system.
I go to New York a few times a year, so I am used to taking the train.
Despite living near Philadelphia and visiting often, I will never get used to how expensive it is to park there!