Bridgerton does not explain what happened to Daphne's father, Edmund Bridgerton, but Julia Quinn's books provide insight into his fate. The show makes it clear that he has died, since mother Violet is introduced as his widow and eldest son Anthony has assumed responsibilities as Lord Bridgerton. The father's absence is noticeably felt by all, but especially by Anthony, who struggles to lead by his example. Edmund's death, while not explicitly shown or spoken about in the television series, does receive a proper description in the second book of the Bridgerton series, The Viscount Who Loved Me.
While The Duke and I follows Daphne's romance with Simon Basset, which is recounted in Bridgerton season 1, The Viscount Who Loved Me follows Anthony Bridgerton, who is referred to as a "Rake" by Lady Whistledown, and his courtship of Kate Sheffield. Anthony consistently doubts himself and his duties to his family and society. While he knows he is expected to marry well and produce an heir, he has held off for fear of his own mortality and his inability to become like his father, who was his role model. This fear stems directly from his father's sudden death.
In The Viscount Who Loved Me, Edmund Bridgerton dies from an apparent bee sting. It's a devastating blow to the family, since Edmund is only 39 and in seemingly good health when he passes. Anthony can hardly believe it himself, since he and his father were both stung when Anthony was younger and they didn't experience any adverse side-effects. The author Julia Quinn says in a closing author's note that, medically speaking, "bee sting allergies generally don’t manifest themselves until the second sting," so it makes sense that Edmund could have suffered a fatal allergic reaction to another bee sting. In the early 19th century, allergy tests and treatments were not yet discovered, so doctors wouldn't have been able to help Edmund regardless.
This experience accounts for Anthony's fear of bees and how he's always checking his watch. It factors into a moment later in the book when Kate Sheffield is pursued by a bee and Anthony overreacts to it. Bee imagery is also prevalent in Bridgerton: a bee circles its nest in the opening credits, it alights on the door knocker of the Bridgerton residence in the first episode, and it's seen on a windowsill after Daphne gives birth to Simon's son. It's a nice little nod to the books and represents that Edmund Bridgerton is still a part of his family's life, even though he is not there physically.
The bee may also signify what's to come in Bridgerton. While it hasn't been confirmed yet, the second season will likely follow the events of the second book and its focus on Anthony's love life. Perhaps viewers will also get to see a live-action version of the late Bridgerton patriarch and his influence on Anthony and the rest of his children before his untimely death.