To get us started, let's make sure we are on the same page. Hyperbole is a literary device that uses exaggerated language to get a point across. These hyperbolic claims are not meant to be taken literally. For example, if I said, "that dude was 100 feet tall," I don't actually mean he was 100 feet tall; I simply want to make the point that the guy I saw was taller than most people.
In "Sonnet 130," Shakespeare is making fun of the common use of a sonnet which is usually to express love, devotion, or sorrow. Here, he is using the poetic structure to make fun of his lover. Most of the claims he makes are hyperbolic because he is making the point that figurative language as such does not and cannot capture the reality of love.
For example, he says his "mistress' eyes are nothing like the sun," poking fun at the fact that her eyes are not actually glowing, bright, and or beautiful. He also says her breath "reeks" and her "breasts are dun" to show that she is the opposite of what is considered beautiful. She may not be the stereotypical fair as a rose, but to suggest that she is ugly, pale, or smells is an exaggeration.