This quote takes place at the end of Act 1, scene 5. Romeo and Juliet have just fallen in love, essentially at first sight. Taken with each other, they haven't even exchanged names, choosing instead to flirt and share a kiss at the Capulet party. When their romantic moment is cut short by the Nurse, Juliet heeds her mother's call and leaves. Romeo takes this opportunity to find out more about the woman he has fallen in love with. When he asks after the identity of Juliet's mother, the Nurse tells him that she is the "lady of the house" and that any man lucky enough to marry Juliet "...shall have the chinks." (1.5.113-117)
At this news, Romeo's heart sinks. He says in an aside, "Is she a Capulet? O dear account! My life is my foe’s debt." (1.5.118) Realizing that she is a direct descendent of his own family's mortal enemy, he makes mention of the price of love in his use of the word "account". This indicates that his life is no longer his own, but rather, Juliet's. In other words, Romeo's enemy has control of his life from this point on.