Why is the verb "dust" used in opposite forms?

The reason it's even possible for to dust to have two contradictory meanings is simply that there's no single fixed relationship between the underlying noun and one particular activity closely associated with it (logically, one might be just as likely to apply dust as to remove it).

Much the same applies to some other nouns used as verbs...

seed (seed a lawn = apply lawnseed, seed a tomato = remove the seeds)
stone (stone a heathen = throw stones at him, stone a cherry = remove the stone)
screen (screen a movie = present on a screen, screen from view = hide behind a screen)
etc., etc. _{(more examples on mentalfloss.com)}

Thus, to the extent that there's a reason for the dual use, it's because in the real world there are multiple relatively common actions associated with the base noun, and in most cases context makes the relevant meaning obvious, so this doesn't cause problems for native speakers.

It happened the same way all language happens: somebody needed a way to say something, and found a way. In this case, different people (well, maybe the same person, but probably different people) needed ways to say "put dust on" and "remove dust" in different contexts, and both used the obvious (in English) method of verbing a noun.

It probably never occurred to either of them that it might be ambiguous, because in context, it wasn't. In most contexts, it still isn't.

You can tell the meaning of to dust only by the context where it is used. What do you do with Swiffer? You clean the dust. What do you do to a dresser with a bag of confetti? You can't remove dust with it. That's the way English developed over the years with contronyms because you don't need to coin or borrow other verbs as they can never cause any confusion.

There are many contronyms in English as the link indicates. To dust is one of them as it means:

To add fine particles, or to remove them

The linked Wikipedia article calls it auto-antonym:

The terms "autantonym" and "contronym" were coined by Joseph T. Shipley in 1960 and Jack Herring in 1962, respectively. Some pairs of contronyms are true homographs, i.e., distinct words with different etymology which happen to have the same form. For instance cleave "separate" is from Old English clēofan, while cleave "adhere" is from Old English clifian, which was pronounced differently. This is related to false friends, but false friends do not necessarily contradict.

Online Etymology Dictionary explains that "to rid of dust" comes form the noun dust which meant:

Old English dust, from Proto-Germanic *dunstaz (source also of Old High German tunst "storm, breath," German Dunst "mist, vapor," Danish dyst "milldust," Dutch duist), from PIE *dheu- (1) "dust, smoke, vapor" (source also of Sanskrit dhu- "shake," Latin fumus "smoke").