Why does royalty have to suffer in Shakespearean tragedies like Macbeth?

To answer this question, we need to look at (1) what Shakespeare based his aesthetics upon and at (2) how royalty was perceived in this time period.

Shakespeare's aesthetic was derived from and largely adhered to the essentials of Aristotle's philosophy of drama/poetry (called aesthetics of drama/poetry and usually described in relation to an individual author). Shakespeare's contemporaries, Sidney and Spenser, similarly built their poetic aesthetics upon Aristotle's poetic philosophy as did their forerunner, Chaucer. One point that Aristotle makes about drama/poetry and that Shakespeare adheres to is that drama/poetry needs to feature noble characters who are flawed but not because they are reprehensible, merely because they are human, and who suffer terrible consequences and suffering as a result of their human flaws. Kings, such as Oedipus Rex, Antigone, Macbeth, Lear, are seen as the most noble of characters, and each is flawed in some way of other by human nature and each suffers terribly. (Wordsworth radically altered this aesthetic with his collection of Lyrical Ballads as described in the Preface to Lyrical Ballads.)  

Regarding how royalty was perceived in this time period, remember that this is before the establishment of the democracies that are so important in today's culture. Most countries were ruled by some sort of monarchy. These monarchies were often considered, at least in public, to rule by "divine right." As such, they ruled because God wanted them to rule.

In Macbeth, when Duncan, the proper king, is murdered by Macbeth, the audience feels that right has been thwarted. This creates a desire on the part of the audience to see how the action unfolds—how wrong will be defeated and society restored to its proper (and God-given) order.

Since Macbeth has highly questionable qualifications for the position of king, the audience is even more interested in seeing him defeated. He is not the rightful heir to throne (that would be the king's son, Malcolm). This is an affront to their sense of divine right.

When royalty suffers in Shakespeare's plays, the audience is drawn into the play emotionally, Since royalty is involved, it is though the entire fabric of English society affected, not just one person. The monarchy was such a revered aspect of most people's lives that suffering on their part was probably easily transferred to the audience members. Shakespeare is tapping into what is probably the most important shared aspect of English life in this time.