Falstaff fakes his death for survival. After the Douglas attacked him, and he immediately dropped down and played dead so he would be spared. Then he stabs Hotspur's leg after Hotspur is dead and brags that he killed Hotspur. This is just a typical,comedic instance of Falstaff's self-promoting lying. Remember how he said that he fought a hundred men when it was only two (Hal and his friend) earlier in the play.