“The lady protests too much, methinks.” Explain why this is an example of dramatic irony.

Although we never learn exactly what changes Hamlet asked The Players to make in their script, it is almost certain that the changes included the dumbshow portion that presents the Fellow pouring poison into the King’s ear as well as the lines about fidelity and widowhood. One aspect of the irony is that Hamlet requested the changes so he could watch Claudius’s reaction: to “catch the conscience of a king.” He later tells his step-father that the play’s title is The Mouse-Trap, but his mother’s reaction is initially stronger.
The irony concerning his mother’s remarks relates to the degree of her involvement in the regicide. Hamlet probably meant to similarly trick her into revealing her involvement, or at least shame her into reacting. However, it is never certain whether her reaction incriminates or exonerates her; Gertrude’s reaction can be interpreted as reflecting either her guilty conscience or her lack of insight into the murder plot. While there is some doubt that she was complicit in her husband’s murder, the fact remains that she did marry his brother sooner than was customary, and could not be unaware of public opinion of that impropriety.
Proclaiming her unswerving love for her husband, the Player Queen calls down a curse on herself if she were to remarry if he died: “In second husband let me be accurst!” Before leaving to go to bed, she continues, “. . . never come mischance between us twain!” referring to herself and the Player King. Upon her exit, Hamlet asks his mother how she likes the play, and she utters the line about protesting.
While Gertrude may be complaining about the overacting, the most common interpretation is that she is acknowledging the inverse parallel to her own behavior.
http://shakespeare.mit.edu/hamlet/hamlet.3.2.html

Dramatic irony is a literary device whereby we, the audience, know something that a character in the story doesn't. The line in the question comes from act 3, scene 2 of Hamlet. Hamlet is staging a play called The Murder of Gonzago, which is meant to reflect the murder of his own father by Claudius. The Player Queen, obviously meant to be Gertrude, declares somewhat overdramatically that she will never remarry in the event of her husband's death.
Hamlet is keen to gauge his mother's reaction to her representation on stage. It's then that she utters those immortal words: "The lady doth protest too much, methinks." The dramatic irony lies in the fact that the Player Queen behaves in exactly the same way that Gertrude did in real life, but Gertrude herself is so singularly lacking in self-awareness that she cannot see this. Or if she has caught on, she is not showing it.