Why is Hamlet's speech to the players important?
This speech is important because it gives us insights into Shakespeare's theory of acting. We will recall that we have no indication outside of the plays of what Shakespeare thought or desired: we have not one personal letter or scrap of a journal entry from him. Therefore, Hamlet's speech provides a good deal of information about how Shakespeare viewed theater.
Suit the action to the word, the word to the action, with this special observance that you o'erstep not the modesty of nature . . .
Clearly (as can be seen as well in plays like A Midsummer Night's Dream), Shakespeare had problems with the actors who "hammed" it up too much. Hamlet also instructs the actors that their goal is to please the discerning theatergoer. Doing otherwise might make the audience laugh, but it does no good in the long run:
Now this overdone or come tardy off, though it make the unskillful laugh, cannot but make the judicious grieve, the censure of the which one must in your allowance o'erweigh a whole theatre of others.
Hamlet says all this because he wants his play to be as realistic as possible so as to trap Claudius into an honest and heartfelt reaction. However, it does reveal much about Shakespeare's thinking as well.
Shakespeare is known for inserting a play within a play, and he certainly puts the form to good use in Hamlet. I assume the speech to which you refer is in Act III scene ii, after Hamlet has berated himself for failing to conjure up the passion to follow through with his plan to avenge his father. Here he gives the players some advice, reminding them not to overact. Instead, they are to see their job as holding "mirror up to nature"and presenting a realistic and believable story.
This speech is important for several reasons. First, Hamlet demonstrates his acting experience. One of the central questions of the play is whether or not Hamlet is mad or "mad in craft." After he meets the Ghost, he tells his friends he may be acting mad; and in this speech we learn his view of and expertise in acting. Second, Hamlet is eager for the play to help him "catch the conscience" of Claudius. If the story is told in a less than realistic manner, he cannot be sure of a true response from the King. Hamlet speaks, the Players listen, and the King reacts.
In his speech to the players in act 3, scene 2, Hamlet urges them to keep it believable, to make their performance as naturalistic as possible. Hamlet intends to use the play—The Murder of Gonzago—to make Claudius squirm and so that Hamlet can gauge his reaction to a dramatic reconstruction of the murder he committed. This will only happen if the actors approach the performance in the correct manner, and that's why Hamlet instructs them on how they should act. Hamlet wants to make absolutely sure that when Claudius watches the play he's given an uncomfortable reminder of the foul deed he committed and the consequences that must follow from his actions. The play is an important component of Hamlet's revenge plan, and so it's absolutely essential that the actors play their part accordingly.