Why does the "monster" come alive in chapter 20?Frankenstein by Mary Shelley

The creature has been alive since early in the book on that fateful November day when Victor brought him to life and then abandoned him in the apartment.  However, it is not until meeting the DeLacey family that he is introduced to and truly understands love, beauty, and all that he is denied in life because of his hideous appearance that he becomes fueled by thoughts of revenge and anger toward his "father," Victor Frankenstein. When the creature meets with Victor in the mountains and begins telling his tale in Chapter 11, Victor has fleeting moments of humanity and tenderness toward the creature who murdered his brother William and frames poor Justine for the murder.  It is during this visit and the creature's logical, articulate and emotionally charged argument that Victor agrees to create a companion for his "child" so that he will no longer be alone in a world which will not accept him.

In Chapter 20, Victor's fears get the best of him, and rips apart the female creature that he has begun work upon.  He has come to the realization that he made a horrible error in playing God and creating the creature, and that he will not make the same error twice.  Victor is, for the first time in the novel, prepared to fully and completely take responsibility for his actions instead of retreating into some fever or illness or a hike in the woods.  It is as a result of this action--the destroying of the female creature--that the male creature's revenge is ignited and his declaration of revenge is unparalleled.  He, in turn, destroys every person Victor loves, therefore, creating a life-long companion for himself...Victor.  Both of them live for the other in their unhealthy relationship.  The creature cares of Victor by leaving him hints, food, supplies.  Victor continues pursuing the creature even when it means imminent death due to his human frailty.  Only after Victor is dead, does the creature truly have no other reason to live, and he jumps out of the ships window after bidding his "father" goodbye one last time.

So, the simple answer to your question is revenge.  The creature truly comes to life in Chapter 20 as a result of his desire for revenge on Victor Frankenstein.