What does “Jus’ folks is jus’ dumb” mean?

"Jus' folks" is a rural-American dialectical expression meaning, roughly, "plain, ordinary people."

Saying

"Don't make the mistake of thinking that jus' folks is jus' dumb"

means don't think people are stupid just because they don't speak in an educated style (so they don't appear to be very smart).

As for the disagreement in number, that is itself a hallmark of rustic speech:

We was nigh on to sunrise before the dog come home.

(Note also the use of present tense to indicate past tense in that rustic colloquialism.)

In both cases, "jus'" is short for "just". It's intended to mimic the dialect of "ordinary" people.

The phrase "jus' folks" is intended to refer to those same "ordinary" people.

In the phrase "jus' dumb" the "just" is an adverb meaning either "only" or "very" (or maybe both), modifying "dumb", which is being used as an adjective.

So, translated:

"Don't make the mistake of thinking that ordinary people are merely dumb."