Why should we care about the Empress?
Because she represents Burnham's failure. Here was the woman that not only did she betray, but also got killed. The Empress is a living embodiment of that failure, that Burnham has demonstrated that whether it is a good idea or not, she will not let her die again. She is a living representation of Burnham's guilty conscience and an opportunity to drive Burnham's sense of duty to Starfleet against her sense of duty to a representation of her mother figure. It is an opportunity for character drama.
What was the point of VoqTyler?
Another way to torture Burnham. She has been intimate with the being that ate her Captain, that assisted in the death of her Captain, her mother figure. She let her guard down, which was taught to her by her Vulcan foster father, with him and was completely betrayed. It is an opportunity for character drama.
What was the point of Culber's death?
Two fold... to drive home the betrayal of Burnham (not only did he betray her, but he killed the husband of a man she considers a friend at that point) and to create emotional conflict (in the future, presumably, currently their plates are a bit full) for Stamets. It is an opportunity for character drama.
What was the point of Landry's death?
Two fold - one, to kill one of the people Lorca considered a confidant which makes him draw Burnham closer, and two to show how dangerous the Tartigrade could be.
What is the point of Cadet Tilly on a starship?
She is the heart and comic relief. She is funny, and she holds the same place that in Whedon's works would have been held by Willow, Fred, and Kaylee.
I understand if these are not reason enough for you to watch a show, but as I see it it's mostly about character drama.