In agreement with all the above, perhaps we should take a look at the title as well.

A number of characters in Shakespeare's plays could be called "princes of darkness." Some examples are Hamlet, Macbeth, Coriolanus, Othello... The dark crevices of their hearts and minds were instrumental to the tragic endings that the playwright succeeded in achieving.

By contrast, Shakespeare may be aptly called a "prince of light" because he enlightens audiences of all times about the whys and wherefores of human nature. In fact, the whole range of personality, character and  motivation, as well as the passions and affects that rule behavior can be found in his works. 

Thus, the curtain that he raises is not only the divide between stage and audience, but also one that, until raised, conceals the best and the worst in mankind. 

Shakespeare was able to encapsulate all of human emotion in his plays, for audiences of all levels to contemplate: think 'all the world's a stage...' He was responsible for highlighting social and moral injustice, historical and present day idiosyncracies all whilst broadening the vocabulary of the masses. He is able to illuminate all who are open to the depths of his work.

The idiom "raising the curtain" means to bring something to clear understanding; to remove any secrets or vague notions that block understanding. This is appropriate for Shakespeare because he was in the business of providing clear understanding of the nature of life and humanity; of the universal questions that relate to being human. One example is his revealing portrayal of women in plays like As You Like It, with Rosalind, and Much Ado About Nothing, with Hero.

I agree with post #2, the curtain is a metaphor for Shakespeare's connection to the stage and performance.  By 'raising the curtain,' Neruda possibly suggests the idea of letting the performance begin; it is the physical manifestation of revealing the playwright's thoughts and emotions to the audience.

The most obvious response seems to be Shakespeare's relation to the theater. Using the curtain metaphor is suggestive of the nature of some of Shakespeare's work (plays) and suggestive of the idea that the Neruda's attempt is to look past the artifice of the work and get "backstage".