In the play's "Induction" (prologue) a figure wearing a robe "painted full of tongues" steps onto the stage. This figure is not a human character – it's a personification of rumor or, hearsay – the kinds of stories that are circulated without any confirmation or certainty. In other words, Shakespeare takes an abstract concept, rumor, and gives it human characteristics. ("Rumour" is typically played by an actor who also has another role in the play though, we've seen productions where a bunch of actors come on stage – wearing scary costumes "painted full of tongues" – and deliver Rumour's speech in unison.)
OK, now that we know what Rumour is, we need to think about what Rumour does and why it matters. Rumour says it's come to "stuff" men's ears full of lies about the recent war between the king's forces and the rebel army. That's just what happens in the play's opening scene, where Northumberland receives contradictory information about the outcome of the battle at Shrewsbury.
Rumour also tips us off that there's going to be a whole lot of outright deception and lying up in this play – Falstaff's swindling of Mistress Quickly, Prince John's deception of the rebel leaders at Gaultree Forest, and so on.
We also want to point out that Prince Hal, who has created a complete persona or disguise ("wild Prince Hal") has prompted the entire kingdom to gossip and speculate about the kind of king he will become when he inherits the throne. When Hal finally reveals his true nature in the play's final scenes, he "mock[s] the expectation[s] of the world" and defies all the rumors and speculation about his character.