The Victorians were deeply concerned with character, with propriety, and with virtue. They were publicly committed to doing the right thing in a visible fashion; they wanted to be good, and to have their appearances match their character.
Jekyll and Hyde turned all those desires on their heads. It told them, "You don't want to be good. You really want to be bad—really bad!" It told them, "You can't trust someone based on appearances. Those change. You all have secrets."
It also addressed a fear of science. Science was engaged in overturning all that was precious to Victorians. There are lots of examples, but the most specific example is the theory of evolution that emerged during this time. Victorians wanted to be on the side of angels; evolution said, "You're essentially beasts!" So did Jekyll and Hyde.
To go along with the other posts, Dr. Jekyll essentially reduces the all-important, Victorian-revered soul to a science project. All he needed to do was find the proper ingredients and suddenly he had the key to controlling vice and virtue, the two qualities Victorians saw as being governed by the soul. Another element that would have clashed with a "respectable" Victorian audience was the blatant reference to Dr. Jekyll's taste for things not befitting his social status, and Hyde's complete abandon when prowling the streets. Before the book was even published, Robert Louis Stevenson was advised by his wife and others to edit the story so that it would not be so shocking to his readers, so the references to vice in the finished product the Victorians received was actually mild compared to what Stevenson initially wished to include.
many victorian men had double lives, during the day they could be a high end banker but at night many men hung around brothels smoking pot. as manyof these men read the book they would begin to feel worried "will there secret life become exposed". religion also played a large part in the book and on victorian society, many people strongly believed in god so he idea of someone meddling with god and changing his design of the human race deeply horrified the victorian reader, also in the book hyde was described to have satans signature upon his face.
ihope this helps