Shakespeare is considered the greatest playwright in the world. This includes not just English-speaking countries, but everywhere. There are a number of reasons for this, and these explain why we keep studying "the bard."
Admittedly, Shakespeare's language can be a hurdle, but today, there are more and more tools to help understand what he is saying, and if you can get beyond the difficult wording, you can start to delight in his work.
Here are some reasons we keep studying him:
Characterization: Shakespeare seemed to intensely feel what his characters felt, whether they were good or evil. He created characters that are powerful because they seem so alive. Their struggles and emotions are completely heartfelt. They behave consistently, as if they are real people. They draw us in with their deeply felt desires.
Language: Shakespeare doesn't simply put flat words into his character's mouths. The words he uses are so memorable that they stick with us. He's never afraid of a pun or a metaphor. Juliet, for example, is not simply beautiful: she sparkles like a jewel against an ear.
Range of plays: Not every play Shakespeare wrote was a masterpiece, but he wrote a large body of plays, and very many of them are masterpieces. He excelled at both comedy and tragedy and did so numerous times. The consistency of his excellence is stunning.
Influence on literary culture: Allusions to Shakespeare show up everywhere, from popular writers like Agatha Christie and P.G. Wodehouse to more literary writers like Aldous Huxley and Tom Stoppard. It's good to study Shakespeare because he is a window into so many writers after him.
Exciting plots: Shakespeare is seldom boring: he brings us witches, ghosts, fairies, laughter, fighting, horror, romance, and bloodshed.
For myself, I have grown to love Shakespeare more and more because he is a humane writer with a moral compass. He understands what it is to be the underdog, the importance of mercy in people with power, and what suffering is. He always stands up for decent values. To me, these are perhaps the most important reasons to study him.