Yes, a reader of Wide Sargasso Sea who is unaware of its connection to Jane Eyre is pretty much guaranteed to have missed the entire point of the novel.
Jane Eyre is so iconic that Rhys could simply assume knowledge of its plot while writing her own book. The heroine of Wide Sargasso Sea, her husband, and her half-brother are all very important characters in Brontë's novel. Without prior knowledge of those characters as they are in Jane Eyre, one would not get much out of Rhys's work. The power of Wide Sargasso Sea is that it forces us not merely to rethink where our sympathies lie vis-à-vis those characters in Brontë, but also to re-evaluate the entire world view that Brontë as well as her readers (ourselves included) take unquestioningly for granted.
So closely does Rhys's novel depend on Brontë's that there's no way to discuss their relationship without spoilers. Spoiler alert: the following reveals important plot points about both Jane Eyre and Wide Sargasso Sea.
In Jane Eyre, the eponymous heroine falls in love with her employer, a rich man named Mr Rochester. She is a governess in his household. She has noticed some mysterious goings-on: strange laughter, an unexplained murderous attack on one of Rochester's guests, etc. But she accepts the stated explanation that a drunken servant named Grace Poole is responsible for these events.
When Jane and Rochester are in the church about to be married, it is revealed, shockingly, that Rochester in fact already has a wife. She is a rich Caribbean heiress named Bertha Mason, and Rochester says he was "tricked" by his father into marrying her for her money. But as his father knew, Bertha is mad. So Rochester has kept his wife shut up in the attic. Grace Poole is in fact the servant who looks after Bertha, and the latter is the one responsible for the mysterious incidents.
Brontë's novel has no sympathy for Bertha. She is simply the inconvenient impediment that keeps Rochester and Jane apart. She cannot be a true wife to him, but prevents him from being able to marry the woman he truly loves. Ultimately she is responsible for burning down his house, killing herself and grievously injuring him in the process.
Indeed, Rochester is supposed to be the sympathetic figure. Poor man, tied to a mad wife! Thwarted in his true love because of a marriage he was tricked into! Nice enough to keep the lunatic at home instead of dumping her in an asylum! Punished for his attempted bigamy by being blinded and rendered lame while attempting to rescue the cray-cray bitch from a fire she set herself!
Wide Sargasso Sea neatly inverts these sympathies by showing us Bertha's point of view and telling her back story. Despised by the English for being a colonial; shut out of Jamaican society by virtue of being white and identifying with the English; married off to a cold and manipulative Englishman who carts her off to a cold and lonely island. Her husband becomes tremendously rich because of her money, then locks her up in the attic with nobody but a drunken servant for company; he then proceeds to have a love affair with the governess and is on the point of bigamy.
Jane Eyre is popularly regarded as a feminist heroine because of her independent-mindedness, but Rhys shows that the feminist independence of Jane, and the eventual triumph of her "true love" with Rochester, depend on the unacknowledged suffering of those who were exploited by Victorian imperialism and mercantilism. Being from the West Indies herself, she saw the violence inherent in the vast mechanisms of trade. The mercantile wealth of Victorian England was built on various sorts of exploitation: slavery, colonialism, sexism. Rochester's wealth comes from his wife, who in turn inherits it from her stepfather, who ran a plantation in Jamaica where slaves worked. Jane too inherits money from a distant relative who has become rich in the West Indies. The source of this tremendous wealth goes unexamined in Brontë, because imperial exploitation was an unremarkable fact in the Victorian world-view. But by retelling the well-worn narrative from a fresh perspective, Rhys opens our eyes and causes us to reassess our values.
tl;dr Read Jane Eyre, then re-read Wide Sargasso Sea to get what the latter novel is really about.