How is the Nurse responsible of arranging the illegal marriage of Romeo and Juliet?

The nurse acts as a go-between for Romeo and Juliet's relationship and marriage.

The nurse is the person who first tells Romeo who Juliet is, and who first tells Juliet who Romeo is.  Therefore, she is the one person who is aware from the beginning that Romeo and Juliet are attracted to each other.  She really does nothing to dissuade either of them, except to tell them the family name of the person.  That is not enough.

After Romeo and Juliet have their secret talk in the orchard, the nurse acts as the go-between for the two young lovers.  Romeo makes an arrangement for Friar Laurence to perform the marriage, but it is the nurse who makes it possible by carrying the message for the two of them. 

Nurse

Good heart, and, i' faith, I will tell her as much:
Lord, Lord, she will be a joyful woman.

ROMEO

What wilt thou tell her, nurse? thou dost not mark me.

Nurse

I will tell her, sir, that you do protest; which, as
I take it, is a gentlemanlike offer. (Act 2, Scene 4)

Romeo and Juliet might have gotten married without this message, but Romeo needed a way to get a message to Juliet so that she would know to go to Friar Laurence.  She would pretend to be going to shrift when in fact she was getting married.  Romeo then used a rope ladder (“cords made like a tackled stair”) to sneak in with, which he provided to the nurse ahead of time.

Even after Tybalt’s death and Romeo’s banishment, the nurse takes an active role.  She chides Juliet for still caring about Romeo after he killed her cousin.  Then she chides Romeo for whining about being banished.

Nurse

O, he is even in my mistress' case,
Just in her case! O woful sympathy!
Piteous predicament! Even so lies she,
Blubbering and weeping, weeping and blubbering.
Stand up, stand up; stand, and you be a man:
For Juliet's sake, for her sake, rise and stand …(Act 3, Scene 3) 

The nurse is thus the only one other than Friar Laurence who knows that Juliet is married when Romeo is banished.  Juliet is horrified, of course.  The nurse urges her to get over it.  She tells her that she can put the whole mess behind her by marrying Paris, and that Romeo is a “dishcloth” compared to Paris.  Unfortunately, by now it is too late.  Juliet is set in her ways.  She has already married Romeo.

The nurse's intervention in helping Juliet get married was an act of love.  She raised Juliet from a baby, literally nursing her.  Juliet was in love, and it seemed silly to let something like a feud get in the way of young love.  She let her affection for Juliet cloud her judgement, and it led to the death of her charge and Romeo.