It's a question asked by many an incredulous school student, to be sure. To an extent it's more one for r/AskLiteraryStudies but I'll try and give you a historical perspective as much as I can too.

There are a few things going on here, so let's take them one at a time in no particular order...

People don't talk in scripts like they do in real life

This is as true of modern movies and TV as it was in Shakespeare's day, or in Ancient Greek theatre. In real life, people talk over each other, they mis-speak and stutter or stammer, they try and finish each others' sentences, they go off on tangents. There's a great video by the excellent "Nerdwriter" on YouTube about how a particular director does it well, but even then it's still slicker and more polished. Even the most "realistic" dialogue on film/TV is edited and presented in a way it isn't IRL.

In fact, one of the ways that opponents of George W Bush during his presidencey used to portray him as stupid was simply to produce verbatim transcripts of his off-the-cuff comments, leaving in all the ums and ers, all the slight mis-speaks and stutters. Because we're so used to seeing polished, perfect dialogue on screen, it made him look like practically incoherent.

OK, so it's unlikely that people in Elizabethan England spoke exactly like the characters in Shakespeare's plays.

The "flowery" language was part of the attraction

Quentin Tarantino is a director who gets a lot of praise for his dialogue, and it's not because it's realistic - in fact quite the opposite. It's stylish, it's witty and we all wish we could talk like that. It's how we show James Bond being cool under pressure, or Jerry Maguire putting aside the smooth-talking persona and laying himself open.

Francis Meres, in a book published in 1598, lists Shakespeare among the great poets and particularly praises his deft ability with language:

"As the Greeke tongue is made famous and eloquent by Homer, Hesiod, Euripides, Aeschilus, Sophocles, Pindarus, Phocylides and Aristophanes; [...] so the English tongue is mightily enriched and gorgeouslie invested in rare ornaments and resplendent abiliments by Sir Philip Sidney, Spencer, Daniel, Drayton, Warner, Shakespeare, Marlow and Chapman

Other contemporary mentions of Shakespeare plays often talk about the excellent speeches and wordplay alongside things like impressive sets and costumes. So it was clearly an attraction for at least part of the audience.

At the time - and still to this day in some prestigious institutions - a grounding in "The Classics" of ancient Roman and Greek myth and history was considered an essential of any rounded general education. So when Shakespeare has his characters talk in classical imagery, he's not (just) showing off, he's speaking a language that the educated would understand and appreciate. The Royal Shakespeare Company has a rather good page talking about some of them.

Not even everyone in Shakespeare talks like that

Shakespeare wrote for a number of different audiences - some plays were written for the general public, others were private commissions for events at court which were later transferred to the stage. Because records kept were not specific a lot of the time, there is an entire branch of scholarship devoted to working out the dates and other details of the writing and performing of Shakespeare's plays - pick up pretty much any copy of a single play and you'll find a whole chapter on the subject.

Characters in Shakespeare tend to speak in either verse or prose, and while it's not a hard and fast rule, you tend to find the peasants and comedy characters speaking in prose and the main characters and gentle/knightly/noble/royal people speaking verse. Again, this is a stylistic choice designed to appeal to both the well-educated elites who would commission him and the ordinary folk who flocked to see the public shows. It's the reason why even the most serious tragedy like Macbeth often has a funny bit with a rude porter to get the groundlings laughing.

An analogy I like to make for a modern production doing a lot of what Shakespeare did is Hamilton: An American Musical. The writing is highly praised for its weaving in references that different parts of the audience "get" - some to classic hip-hop and rap, some to older musicals, some to history, some to previous works by the author and so on. Language is used to demonstrate character (Lafayette begins by getting his rhymes wrong and being unable to think of words, and ends with one of the fastest raps on record). Nobody actually talks like Hamilton in their day-to-day life but it's a heightened form of language that is understood by everyone.

All that is fine, but what's with all this thee and thou business

Shakespeare was writing at a time of change in the English Language.