Two reasons for the decline of Elizabethean theatre is the closing of the theatres by the plague outbreaks during 1592 caused theatres to close for 2 years. After that, the Puritan's closing of the theatres in 1642 as immoral and frivolous certainly caused the theatre to suffer a black eye.
The Jacobeans and those who followed did not have Shakespeare's mastery of the human spirit and failed to understand that at the core of human personality is a deeply moral drive. The Jacobean tragedy was darker and much more focused on revenge. The style reverted back to the Greek tragedy of Seneca. Even in Shakespeare's tragedies, the moral triumph of law remained a firm constant. Shakespeare was a genius in unravelling the deepest secrets of the human spirit.
In comedies, the theatre degenerated into depravity which further alienated the Puritan controlled government officials and encouraged further censorship of performances.
After the death of Shakespeare, the allure of drama gave way to newer genres though not entirely and not all at once. Printing techniques were being perfected and more books were printed to a (slightly) more literate society making the need for the drama a little less noticeable. Also England, in the century following Shakespeare's death, fought several wars with quite a few countries depleting many things needed for the theater: actors, leisure time and an appreciation for the arts all took a hit.
This was happening all over Europe and not all of the decline can be attributed to the death of Shakespeare. However, whenever the master quits producing, often the art form dies- at least in the worst case scenario. Sometimes, it is re-invented and transformed.