What is the meaning of Falstaff's motto "the better part of valor is discretion" in Henry IV, Part One?
Falstaff's motto is essentially a proverb: a word of wisdom that is meant to be used in everyday situations. For example, the proverb "You draw more flies with honey than vinegar" means that it is easier to win people over if you are kind instead of rude. In the same way, the phrase "the better part of valor is discretion" means that the best sort of courage--the kind that constitutes true bravery instead of recklessness--is courage that is guided by wisdom (discretion). Bravery applied blindly without caution and wisdom is meaningless.
Translated to modern english to be "Discretion is the better part of valor", this phrase redeems himself from a cowardly act. He had once become a bragging and proud knight who had cheated by his despicable tactics by cheating death by pretending to be dead to deceive the enemies in order to escape Henry IV's clutches and be killed by him.
This sentence implies that "honour" and "valor" would get you nowhere and he tries to hide his "counterfeiting" by saying that "discretion" is needed so that people won't be foolishly running into a pile of swords aiming right on top of his head all due to honor and reputation, so that he doesn't disgrace himself by playing chicken and running away, which would be very risky and dangerous