Here, Hamlet explains to Horatio the reason he is willing to engage in the fencing match with Laertes. Earlier Hamlet has said "We defy augury." He means that omens and signs, indicating the match may be a bad idea, mean nothing to him, have no influence on him; he has no fear of them.Augury is the divination of, the reading of, signs, omens, portents that indicate the future.
"Providence" has several senses of meaning. In this sense, "there's a special / providence in the fall of a sparrow," Hamlet uses it to mean a demonstration of divine care, direction, concern. He expresses the idea that Providence (God) cares about every evil that occurs and every ill that comes to pass. He means to reassure Horatio about his fate should the duel prove treachery since Horatio has just warned him that he cannot win and that he must listen to his own mind about it: If he has misgivings, he must opt out of the duel.
You will lose this wager, my lord.
If your mind dislike any thing, obey it: I will
forestall their repair hither, and say you are not
Hamlet uses an allusion to a Bible verse in Matthew that declares God controls the life of a sparrow, thus is in control of all even greater lives. Hamlet reveals his belief that his life and future are cared for by God and that, therefore, he is willing to proceed without fear even if his heart is troubled by it.
But thou wouldst not think how ill all's here
about my heart: but it is no matter.