It appears to be because of what they're the conjugate base or acid of. So you can just use the rules,
The conjugate base of a strong acid is an extremely weak base,
And the conugate base of a weak acid is a weak base,
and same way with acid/base switched.
The dotpoint doesn't mean we have to memorise which anions/cations are acidic/basic. (...does it?)
"acidic anions come from polyprotic acids (why?!) "
Well, polyprotic is > 1 H+, so say... Sulfuric acid (most obvious example, but still)
H2SO4 --> H+ + HSO4- so there's an anion that's still acidic, so when an acid gives up a single H+ and it's still got some left, it's an anion that's acidic.
Neutral ions come from strong acid and bases because they're things like... I dunno, Cl- is the conjugate base of HCl, and applying the rule "the conjugate base of a strong acid is an extremely weak base", so Cl- will be an exteremely weak base, ie. effectively neutral.
Next part... Basic anions coming from weak acids, again we use the rule "the conjugate base of a weak acid is a weak base", like, maybe Acetic acid, it gives the acetate ion, which is a weak base that's basic enough to make a solution... basic. Yeah (it's 2am.)
"acidici cations from weak bases" That'd be from a reaction between a strong acid and a weak base, giving an extremely weak conjugate base and a weak conjugate acid, so the salt has pH < 7...