I presume you mean Anne is pregnant [with who would turn out to be Elizabeth].
Both Boleyn sisters had children - Mary, Catherine & Henry Carey; Anne, Elizabeth I - all of whom may have been fathered by Henry VIII. In those days it was not so much who was the biological father as who was officially acknowledged as such. Elizabeth was officially recognised as Henry VIII's child, Catherine & Henry Carey were not. Elizabeth was Anne's first and only surviving child, so the 'first-born' being referred to must be Mary [Mary I] who was daughter of Catherine of Aragon.
In Tudor times, a king needed a son. With a son, he had an heir. A daughter just 'wasn't good enough'. Daughters were inconsequential & any legitimate son born after daughters would automatically become heir rather than the daughters. England had never had a Queen regnant [one who ruled in her own right rather than as wife of a King.] It's quite a common theme in Tudor movies that Henry VIII's overarching desire is for a son who will inherit the throne. [Henry had an acknowledged illegitimate son at this time, but he is not important enough to history for this to be needed for a movie plot].
Henry is always seen as one who will move from affection to affection at whim, though what he needs for an unchallenged heir is a son in wedlock.
At this time, all pregnancies were precarious. Loss of the infant during term or stillborn was more common than children surviving beyond infancy. It was not uncommon for the mother to also die giving birth. Medicine was still at the leeches & bloodletting stage. Disease was thought to be caused by 'bad humours' [smells - microbiology was centuries away].
Catherine of Argagon had six pregnancies, four died at or shortly after birth. Her first to survive beyond that died a mere two months later. Only Mary survived to adulthood.
Anne had four pregnancies. Elizabeth, the first of these, survived. The others did not.
So, historically, had Anne given Henry VIII a son, that would have strengthened her position in court. She would be mother of the undisputed heir to the throne - Catherine's daughter Mary [Mary I] having been stripped of her position following Henry VIII's annulment of his marriage to Catherine of Aragon.
If it were to be a girl, her position would be weakened. Henry VIII may choose to move on to another woman. No-one at the time knew he would go on to marry 6 times, but the plot of these movies is always aware that the audience knows, and knows kings have many mistresses; Henry was not unique in this. They will also know Anne's tragic end. One thing not common knowledge is Mary Boleyn's end, so they can play with that a little. She was in debt after her husband died, banished from court after marrying again, ran out of money & lived in near poverty until her death in her early 40s.
So they can use this foreknowledge as portent in the plot, making it simpler to point to an event & mutter - 'if you do that, bad things may happen' - which often turned out to be true for anyone who fell into Henry's bad graces.
There is also the commonly-used Tudor trope that "everybody is constantly plotting everyone else's downfall". This may be close to actually being true - it appears to have been quite a cut-throat time; courtiers of greater & lesser importance constantly struggling to get themselves closer to the King, which in turn will give them greater influence over any more distant. The King was an absolute monarch at this time. His word was law in most things; which leads to another trope of him becoming incensed whenever he couldn't have his own way. [vis, Pope & marriage annulment, leading to the formation of an entire breakaway church… but that's another story entirely.]
In movie shorthand…. son good news, daughter bad news.
Anne had a daughter, Elizabeth I… bad news.
Henry did later have a son [by Jane Seymour, his fourth wife] who became King Edward VI, but died aged 15, when he was succeeded by, in rather rapid succession, Lady Jane Grey, Mary I then Elizabeth I.
Mary I was the first undisputed Queen Regnant of England.
In all honesty, it's no wonder anyone struggles to follow this plot, because it is about one of the most thoroughly documented Renaissance monarchies & the true historical plot is broader than it is long. Even though I know the general outline, I have to look this stuff up every time ;)
Though the movie has some historical inaccuracies, it's kind of 'close enough'. If you want to learn more about this period [and, boy, is there a lot to read;) start at the Wikipedia page for the movie itself, then all cross-references from there go to each real historical character.