This is a nearly impossible question to answer usefully. Why is anything popular? Because a lot of people like it. Why did so many people like "The Talons of Weng-Chiang"? Here, I need to resort to some guesswork. First, it had Leela, with her uncivilised background, in Victorian London, with its highly mannered society and restrictive attitudes to women. She was so completely out of place that there was ample opportunity for humour, mostly at the expense of the Victorians. Second, there was the "Sherlock Holmes" atmosphere and the Doctor even wore a deerstalker hat. Third, there were some very good guest characters, Jago and Litefoot, who later featured in their own audio series. Fourth, it had some good action scenes, such as Leela jumping through a window in pursuit of villains. Fifth (and extremely important), it had a good story. Sixth, it had memorable villains, including the "Peking homunculus" posing as a ventriloquist's dummy. Other people can probably point to other factors that helped to make it popular. However, the best way to answer the question for yourself is to watch the thing. If you enjoy it, that will give you a good idea why others also enjoyed it. If you don't enjoy it, which is a matter of taste, not of right and wrong, you'll probably just need to accept that it works for others but not for you.
As a general observation, there seems to be something about the combination of Victorian society and science fiction that has special appeal. This showed in Doctor Who most recently with the popularity of Madame Vastra and her companion Jenny in "A Good Man Goes to War". It showed also in Torchwood Series 2, Episode 12, Fragments in the scenes of Captain Jack's early days in the Victorian-era Torchwood Institute. The same thing has been observable in Doctor Who since the 1960s (Evil of the Daleks was partly set in Victorian England). As noted above, Jago and Litefoot, who were not even companions of the Doctor, were popular enough to feature in their own audio series, Jago and Litefoot, which is (I think) now on its fourth series of stories, set in 1890s London.