What is the significance of Romeo's speech (lines 106-113)?
Please note that when you are asking about specific lines from a Shakespearean play, it is very important to name the act and scene number as well. Lines 106-113 appear multiple times (in different acts and scenes) throughout the text.
In looking at the play, there are only two times when Romeo alone is speaking in lines 106-113. As such, I will address both. The first is in Act 1, scene 4, as Mercutio is trying to convince Romeo to attend the Capulet party. In the lines you cited, Romeo explains that his hesitation in attending the party centers around the fact that he has a bad feeling about a "consequence yet hanging in the stars" (line 107). In other words, Romeo has a bad feeling that the party will bring about misfortune in his life. This is significant because it turns out to be accurate foreshadowing. In reality, Romeo will attend the party and will meet Juliet, which in turn will set into motion the events that will bring about Romeo's death just days later.
The second set of 106-113 lines in which Romeo speaks occurs in Act 5, scene 3, the final scene of the play. Here, Romeo addresses what he believes to be both a dead Tybalt and a dead Juliet in the Capulet tomb. Tybalt is, indeed, dead by Romeo's hand. In the lines you cited, Romeo tells Tybalt that he will avenge Tybalt's death by killing Tybalt's killer. Again, this is significant because it heightens the suspense for the reader. After all, Romeo is Tybalt's killer; thus, it becomes clear (if it wasn't before) that Romeo intends to kill himself in the tomb.
Romeo next addresses Juliet, remarking that she is still beautiful in death and that she appears so life-like. This set of lines is significant because, in truth, there is a reason why Juliet appears life-like: She is alive, and is just moments from waking up from the potion she took to fake her own death. Romeo, unaware of this, will drink the poison he bought before she can make the truth known to him.