While the "necessity" of breaking with the Catholic church may be debatable, it certainly appears to have been viewed that way by Henry VIII. The very quick and dirty of the situation is as follows.
Henry VIII's father, Henry VII, was the founder of the dynasty. Having defeated Richard III at Bosworth Field, and soon after having married Elizabeth of York, thus concluding the War of the Roses, Henry VII and Elizabeth of York had seven children, of whom four survived early childhood. They were, in order, Arthur, Margaret, Henry, and Mary.
Arthur, the eldest, was Prince of Wales and heir to the throne. He married Catharine of Aragon (daughter of the King of Aragon and Queen of Castile) when they were in their mid-teens, but he then died a few months later. This made Henry (not yet VIII) the heir-apparent, and only surviving male child who could carry on the Tudor dynasty name.
After Arthur's death, Henry married his brother's widow, Catharine of Aragon. This required some effort, as Catholic laws prohibited doing so. A papal dispensation was received to allow it, in part because Catharine declared that her first marriage had never been consummated (and therefore wasn't valid in the eyes of the church). A couple months after Henry ascended to the throne as Henry VIII, they got married.
Henry VIII and Catharine had at least six children, but some were stillborn and only one lived more than 2 months - a daughter named Mary (the future Queen Mary). With only a daughter living, the Tudor dynasty would become extinct when she married and a new man's family became the line of kings (the dynasty is traditionally based on the man's family, regardless of which side is the source of the authority - this changed for the first time with the current queen, Elizabeth, retaining the Windsor identifier).
Henry VIII was unhappy with this, and as Catharine had reached an age where further children were unlikely, began seeking to have his marriage to Catharine annulled so that he could attempt to have a son with a different woman. Whether he actually believed his own arguments for the annulment from the get-go, used them as a matter of convenience because he wanted to chase after Anne Boleyn, or was genuinely so desperate for an annulment that he came to believe something he had fabricated, I do not know. Equally unknown is the proportion in which his various motivations for pursuing the annulment existed - some egotistical perhaps, to carry on his name, and others more selfless, such as avoiding another succession crisis and yet more war for his country. Either way, he put forward that the bible prohibited marriage to a brother's widow, and that such a union could never produce a son. He argued that his lack of success in this department was proof of this, and that the pope did not have the authority to override scripture. It is possible that he further believed or presented evidence of this by his successful fathering of a son with one of his mistresses, Elizabeth "Bessie" Blount, "proving" that it was not him who was at fault for the lack of a male heir.
The papacy, potentially influenced by Catharine's nephew the Holy Roman Emperor Charles V (who held the pope captive at one point), refused to grant an annulment, and dragged on the proceedings for many years, refusing audiences, sending legates, etc. After 6-8 years of this, Henry finally married Anne Boleyn (secretly at first) without papal permission. He argued that he could not prove Catharine and Arthur hadn't consummated their marriage, so therefore his entire marriage to Catharine had been invalid and he was within his rights to get married without any further input. This was accompanied by arguments that the authority of a monarch in his own lands exceed those of the pope, and that this included spiritual matters. Herein is the break you inquired about - Henry's desire for a son to carry on the family name and dynasty, for the procurement of which he required a new marriage (which the papacy refused to allow), led to Henry declaring his that monarchal authority in England supersedes that of the church hierarchy.
The irony of this of course is that Anne Boleyn didn't give him a son either. She birthed Elizabeth (future Queen Elizabeth) and a couple stillborn children or miscarriages before Henry had her executed in favor of another new bride. This third wife, Jane Seymour, did produce a son named Edward (future King Edward VI) who would be Henry's last child, despite a further three marriages after Jane's death from complications in birthing Edward.
Edward went on to succeed Henry in the now vehemently anti-Catholic monarchy, but when he died young and without issue, he attempted to pass the crown to a protestant cousin (Jane Grey) instead of to his Catholic eldest half-sister, Mary. Mary had none of that, and Jane lasted nine days before Mary seized power at the head of an army. The last of the Tudors to hold power was Elizabeth (after Mary, too, died childless). Elizabeth never married, and the crown passed to the Stuart dynasty after her.