What does it mean when someone says “Don't call me ‘boy’!”

In the given context, Alex Jones is trying to belittle John Oliver by saying him "boy". It is derogatory, if you call a fully grown-up man a "boy".

From Merriam-Webster

an immature male

separate the men from the boys

Also, it is an offensive¹ word for a male servant.

^{1: Check Entry 2 in Merriam Webster}

According to one of my history teachers in high school, whites in the south would call male slaves "boy" regardless of their age... also female slaves "girl" regardless of age. I recall this was done in "Huckleberry Finn", where even kids were expected to call grown men "boy".