Anime is popular for loads of different reasons, and each one applies to a different section of fans.
One reason would be because fans find certain anime characters attractive, from girls who are cute and "moe" to ladies with water melon sized breasts...because of this attraction fan-service packed shows like K-ON, kissxsis, Strike Witches, high school of the dead and many more become really popular.
While other types of anime are popular because they tackles some really compelling subjects in a mature and gripping way. They're literally popular for the same reasons shows like Breaking bad and Game of thrones are! They just contain excellent story telling. (FMA Brother hood, cowboy bepop, perfect blue, tokyo ghoul, spice and wolf, madoka magica, the list goes on)
Another reason is anime is popular is it often goes a step further then any other medium with its action, gore, romance, comedy, everything is just bigger and more out there for the vast majority of shows.
The final reason and probably most obvious reason is it has mass appeal- there is an anime for every gender and age, every lifestyle, every interest. You can find a show that will be to your tastes.