In a sense, this question has it backwards in asking how an Elizabethan play is "relevant" to modern readers. Rather, modern readers should ask what they can learn from reading the play. There are two approaches to answering this.
The first is to assume that certain themes and ideas are universal, and thus the time period in which a work is written is essentially irrelevant to the question of how well Shakespeare deals with such universal themes as love. If that is the case, one can learn from Shakespeare's portraits something about how love functions.
The second approach is to argue that each historical period is unique. Thus, a 21st century reader can only escape the parochial and limited perspective of the 21st century by exploring attitudes of other periods. So, for example, by reading Elizabethan works, one can see how people learned to love each other even in a period of arranged marriages and pre-marital celibacy (at least for upper class women).