Macbeth

Your question could be either "Why do people think Shakespeare's plays are difficult?" or "What are the real reasons people find Shakespeare's plays difficult to understand?" These are quite different questions, because people are often wrong about why they have problems. In particular, they imagine that the plays are written in a different language. This is not true, as anyone will find out if they try to read a translation of the plays into a language they do not know. Assuming that you are not a Polish speaker, try reading a passage from Shakespeare in Polish translation and then look at Shakespeare's words. You will instantly recognize Shakespeare as writing in English; most if not all of the words will be familiar to you. Again, the problem is not the use of unfamiliar words, although Shakespeare does use words which he just made up and more often uses the secondary meanings of words. But actually we humans are well equipped to acquire new vocabulary when reading or listening to a different dialect of our own languages. Americans can understand people from England and even from Scotland without a lot of difficulty if they are willing to try (as they did when the Harry Potter books became popular in the US).

The real reasons for difficulty in understanding Shakespeare's plays are as follows:

1. They are plays, which is to say, they are instructions to actors as to what to say in the performance of the play. They are best understood when they are watched, not when they are read. If you want to know what a movie is all about, do you read the screenplay, or do you watch the movie? Of course you watch it. The ability to read and understand playscripts requires an understanding of dramatic convention and a good imagination, and it is not easy, especially the first time. Unfortunately Shakespeare's plays are usually the first plays students read.
2. Shakespeare wrote using verse. Many of his characters speak in it. What this means is that there is a rhythm to their speech and sometimes Shakespeare bends the syntax to fit the rhythm. So, instead of having Richard III say "buried in the deep bosom of the ocean" he has him say "in the deep bosom of the ocean buried," moving the verb to the end of the sentence. Formerly, this was a common custom among people writing lyrics to songs, to make them fit the rhythm of the melody. More recently, however, lyricists do not try to match their lyrics to the melody; they alter the melody to match the lyric. As a result, English speakers these days have very rigid expectations of syntax and easily get confused when it is altered.
3. Shakespeare often writes long and complex sentences. He is not alone in this, of course, and never reaches the kind of sentences we find in 19th century French writers like Proust and Hugo. But many people nowadays expect sentences to be short and simple. They have never studied the grammar of sentences with multiple clauses. They are not practised in unravelling complex sentences and easily get lost.
4. Shakespeare uses a lot of poetic and rhetorical devices. They are what make his words sing. But song lyrics and narrative prose these days hardly use these devices at all apart from a few simple and chichéd similes, and people rarely read or study poetry like they used to do. As a result many students are unprepared for the poetic richness of the dialogue.