Well, the short answer is that Elizabethans never saw “The Tempest”. Elizabeth had died long before the play was written, and James I was the king at the time, so the English of the day are now called Jacobeans.
“The Tempest” is a fascinating, magical play and there are many reasons why it still appeals to modern audiences as much as it would've appealed to Jacobeans: it's got amazing characters, it's full of magic and strange creatures, it's very funny, it's visually striking and has beautiful language. Jacobeans also enjoyed the stagecraft and special effects possible in the newly developed indoor theater where the play likely premiered: magic tricks, disappearance and other illusions that would have been impossible on the traditional outdoor stage. Royal performances of the time often included a masque, an elaborate fantastical stage show more akin to Cirque du Soleil than a traditional play. They were extremely expensive to put on and they had all of the most wonderful theatrical innovations available at the time. Shakespeare actually includes a masque in “The Tempest” and it would have taken advantage of all the eye-popping effects of the new indoor stage.
The play was also very topical. It’s about travelers being shipwrecked on a magical island and at the time England was starting to explore what they called the New World – we know it today as North and South America. There were all sorts of wild stories coming back from people exploring in Bermuda and the Caribbean about the fantastical people and creatures that they found there. They riveted the Jacobean imagination. It's hard for us to imagine today what it would be like to discover a whole continent full of people and creatures that you didn't know was there before; it was almost like discovering a new planet, but one full of life and one you could travel to. The idea that a European nobleman like Prospero could go and become master of this magical realm played into Jacobean hopes about what a valiant English person could do to conquer the New World; it was fascinating and exciting and a big part of the appeal of the play.