In this scene, Hamlet has just discovered that his father was murdered. He encourages the ghost of his father to reveal the identity of the murderer so that he, Hamlet, can "with wings as swift / As meditation . . . sweep to [his] revenge." What he means here is that he will act and take revenge as quickly as the mind can think. Of course, there is an irony here: Hamlet proves indecisive and hesitant in the play and thinks about how he will take his revenge and whether or not he should for a long time.
Later in the play, in act 3, scene 1, Hamlet reflects on his own indecision and failure to act. He reflects that "the native hue of resolution / Is sicklied o'er with the pale cast of thought," meaning that his resolution, or determination to act, has been clouded and dulled because of his tendency to overthink.
Hamlet does not kill Claudius until the final scene of the play when he has little option but to do so, because he himself has been poisoned. By the end of the play, therefore, Hamlet's promise to his father at the beginning of the play—to take revenge as swiftly as the mind can think—seems nothing but empty bravado.