This is a very good question. Many people have written dramas and plays, and many of them are quite good. So why is Shakespeare so admired, so studied, so loved, so insisted upon?
The short answer is that he was so very, very good at doing what he did, and he did so much of it so well that it really is quite unbelievable. His work is so good that many people do not believe that he and he alone wrote all the plays that are attributed to him, but the fact is that he almost certainly did do so, as hard as it can be to believe when you study Shakespeare.
Some creative people have been so far beyond their own time that they haven't always been completely understood during the years that they lived. Mozart, for instance, was a person like this. His work just sounds finished in a way that other works are not--it's difficult to describe, but even people who don't know much about music recognize that there is something special about what Mozart did. You can feel it.
Shakespeare is the same way. The fact that the language has changed a good deal since Shakespeare's time makes it more difficult for us to see that at first, but with a little help, you can clear away the confusion caused by that to recognize that his work is finished and special in that same way.
For instance, when I teach Macbeth, I spend a lot of time on the beginning of Act 1 Scene 3, when the three Weird Sisters meet in the forest as they had promised to do previously. There's something very special about that little conversation that they have that--if you know what to look for!--you can enjoy it in a way not immediately obvious.
That's what special about Shakespeare. His works are just brimming with fantastic little things here and there to enjoy, but it does require that you know what it is that you're looking at, and for that, you need the guidance of someone who already knows how to do it.
For those of us who like to write--like me--Shakespeare shows what genius can do with words and characters and situations.