Why does the Nurse seek out Romeo?

The Nurse seeks out Romeo at Juliet's behest, as an intermediary. She comes to find Romeo, to give him a warning about abusing Juliet's trust, and basically to see what Romeo's intentions towards Juliet really are:

Pray you, sir, a word: and as I told you, my young lady bade me inquire you out;

Juliet has sent the Nurse to find Romeo, to talk to him and find out more about him. The Nurse has...

The Nurse seeks out Romeo at Juliet's behest, as an intermediary. She comes to find Romeo, to give him a warning about abusing Juliet's trust, and basically to see what Romeo's intentions towards Juliet really are:

Pray you, sir, a word: and as I told you, my young lady bade me inquire you out;

Juliet has sent the Nurse to find Romeo, to talk to him and find out more about him. The Nurse has been taking care of Juliet ever since Juliet was a tiny child, and is very protective of her. So the Nurse also takes this opportunity to issue young Romeo with a gentle threat:

First let me tell ye, if ye should lead her into a fool's paradise, as they say, it were a very gross kind of behavior, as they say: for the gentlewoman is young; and therefore, if you should deal double with her, truly it were an ill thing to be offered to any gentlewoman, and very weak dealing.

To put it in a more modern vernacular, the Nurse is saying: "Listen, mister. If you're leading Juliet on, that's a rotten thing to do, because she is very young. So if you're just two-timing her, that's a terrible thing to do to any lady, and the kind of thing a weak coward would do, besides."

Following this conversation, Romeo delivers a message to Juliet through the Nurse:

Bid her devise some means to come to shrift this afternoon; and there she shall at Friar Laurence' cell be shrived and married. Here is for thy pains.

Romeo tells the Nurse, "Tell Juliet to make up some excuse so she can come to confession this afternoon at Friar Laurence's place. And then not only will he hear her confession, but he will marry us." Then he tries to give the Nurse a coin "for her trouble" in coming to deliver the messages.