Why did Shakespeare use difficult language to write his plays, such as Twelfth Night?

I would argue that Shakespeare's language was not as difficult for his audiences as it is for us. He was writing in what is sometimes called Early Modern English, retaining words, spelling, grammatical forms and pronunciations we don't use today. The language we call Modern English--what is easily recognizable to us today--didn't standardize until around 1700, almost a century after Shakespeare's death.

That said, Shakespeare wrote for his present day audience, not posterity, so he used current slang to maximize laughs and make jokes. Since slang changes rapidly--we might easily be confused by 1960s, 1970s or even 1980s slang--Shakespeare can be difficult.

Also, Shakespeare is read today because his language is so densely packed with vivid imagery, metaphor, and punning. The very richness of his mind can make him a challenge to follow: he was putting complex thoughts and emotions  about death, love, guilt, ambition, hypocrisy and loyalty, to name just a few, into images. Further, Shakespeare, like many famous English authors, including even the ex-Beatle John Lennon, loved punning. This means using words that have more than one meaning in such a way that either meaning could "work." Because the meanings--especially the slang meanings--of many words have changed since Shakespeare's time, the puns can be difficult for us to understand. For example, in Twelfth Night, Act 1, Feste says he has "two points" to make. Points can mean points, as in points in an argument, but also, in those days, referred to the strings or laces which held up a person's pants, so Maria has fun with that, saying that if one "point" in his argument doesn't hold, the other will keep Feste's "gaskins" (pants) up but if both break or fail, his pants will fall down. A contemporary audience would have gotten the joke: we struggle because the language has changed.