Why did Henry VIII break with the Catholic Church?
To be able to divorce Catherine of Aragon and marry Anne Boleyn in the hope his new wife would bear him living male children.
Henry VIII was a Catholic when he began his reign of England in 1509. He supported the Catholic Church's doctrine that the Pope is the head of the church. He made sure all the religious services in England were Catholic and performed in Latin.
Up until he wanted to divorce his (first) wife, Catherine of Aragon, because she had produced no male children who survived childhood (they did have girls who did). He wanted to set Catherine aside and marry Anne Boleyn (Catherine's maid). But Catholic doctrine did not allow for divorce, so the Pope refused to allow the King to divorce Catherine.
So the King broke with the Catholic Church, created his own (the Church of England), and declared that the King was the head of the church. He was then free to divorce Catherine and marry Anne. Who also produced no living male children, and so he executed her 3 years later.
And now let's dive a little deeper into why it is that Henry wasn't granted a divorce from Catherine (thanks to @Lucio Margherita for the additional information!).
Divorce, while probably unheard of among the peasant class, wasn't so unusual for royalty. Think of it as a "scratch my back and I'll scratch yours". The papacy at this time in history was far more than a religious institution - it was also a political force that would issue decrees that the Pope expected would be followed by Catholic kings. But the Reformation under Martin Luther had started and so power was starting to leak away from the Pope as kings and kingdoms started to turn away from Catholicism. So Rome needed kings to follow decrees and the kings would then get favours from Rome.
So along comes Henry VIII, knowing full well all this, and expects there will be no problem getting a divorce from his wife. But it doesn't happen. Why?
One reason could be that Rome was in the process of being sacked.
The Pope, Pope Clement VII, had been watching the political climate in Europe and saw that the Holy Roman Empire was getting strong. Really strong. To try to balance out the power, Clement gave his support to the King of France instead of the Holy Roman Emperor. As you might imagine, that did not go over well with the Emperor and so he attacked France and defeated its army. Which was really good for the Emperor. Except that he couldn't pay for the war he started.
And so there's this army in France, some of whom are from the HRE but a lot are mercenaries and so have loyalty to where their paycheck comes from. Which currently isn't the HRE. Where to go and get paid? Someone pointed out that Rome was relatively undefended and really rich and that sacking and plundering it would be a great idea. Which they did.
Rome was the home of, among other people, the Pope. In order to avoid being killed outright, he was forced to pay huge sums of money and give up large tracts of property. Even with that, he was virtually a prisoner in Rome.
The Emperor was a bit embarrassed by the whole event (not being able to stop his troops from sacking Rome) but Clement was completely traumatized and did everything he could to try to stay in the Emperor's good graces.
And so once again, we take up the case of Henry VIII, trying to set aside Catherine of Aragorn, his first wife, in favour of another woman. Catherine happened to be the Emperor's aunt. And so to avoid upsetting the Emperor, the Pope refused Henry's wish for divorce.