Elizabeth loved the theatre. She would often have companies of actors like The Lord Chamberlain's Men perform at court. In this way she could enjoy the latest play without having to go to the theatre itself. The theatres were located in the red light district and not a proper place for the Queen of England.
It is rumored that she liked the character of Falstaff so much that she asked Shakespeare to write a play about Falstaff in love. The result was one of the two plays with no known source, a play set in a contemporary England, and Falstaff not so much in love as in lust.
A specific type of court entertainment grew out of performing for the queen, this was the court masque. In The Tempest, we see one. Ben Jonson was well know for his masques.
Under Elizabeth, not only theatre but the English language flourished. The term Elizabethan is a direct result of her love of theatre and literature.
Queen Elizabeth adored theatre. In many of the books I've read regarding her life, she considered theatre to be one of her favorite releases from the burdens of her "job"--one of her favorite entertainments. She was one of the earliest patrons of art--theatre included--and this love began even before she was Queen as her father also enjoyed many different artistic ventures. She, like her father, was a truly "Renaissance" woman--many talents, many interests.
Queen Elizabeth was well educated, spoke several languages, and wrote poetry and music. All poets and dramatists of the time paid their respects to the Queen, and Spenser’s Faierie Queene is probably the most elaborate example of that. Elizabeth enjoyed the theater, patronized it, and attended some of Shakespeare’s plays. Although we don’t know for sure all that she attended, we do know she saw Merry Wives of Windsor and Love Labors Lost. Her interest in drama was such that several books have been written that hypothesized that she wrote some of Shakespeare's play, but this of course is only dreamy speculation.