Let's first talk about comics in general. Graphic expression has been part of human culture since neolithic times, so it's not a surprise to find some kind of sequential art tradition in any culture anywhere. Japan's manga culture was just another offshoot of fine arts as it is anywhere else. In every part of the world you can find those moment where high and low culture meet in graphical form. Comics journalism, autobiographical comics, comics have a place for every kind of expression, from the most fantastic to straight nonfiction.
For Japanese youth, manga and anime is just a thing they encounter all the time, everywhere, as I did with comics as a child.There are anime and manga for children which help them learn their language and culture, and promote teamwork and hard work and express typical fantasies such as magical princess and transforming heroes.
If you're a comic reader, you understand that as we get older we look for other stories - stories that reflect our current or future experiences. Manga, particularly, provides that. Girls were*the* driving force of manga popularity in the west, where there is a great deal of content for girls and women, in a way that there is not in superhero comcis.
For western teens, anime and manga provide a secret language - and who doesn't want that? "I was reading the new Yaoi book, did you see it? Hikaru's so tsundere,Yuki's such a boke!" Yeah, Mom has no idea what they just said. Even in Japan, fan terms always run ahead of parental understanding. "She'syandere - moe!" "I dunno, feels nioi-ke to me." No one outside "us" can follow the conversation. When you're a teen, there is enormous power in being "us."
Artistically speaking - we like what we like. Tezuka, the iconic "God of Manga" was moved by Bambi's expressive eyes. Have you watched a Betty Boop cartoon - there's those characteristic big eyes/small mouth. Early Japanese animation and comics was strongly influenced by western animation and comics.
Manga is not one thing anymore than comics is one thing. Read 10 manga that are for different ages, genders and audience, your going to see differences as you would if you read 10 different non-superhero comics. Anime for little girls doesn't really look like anime for adult men.
Personally, for me, it's always the characters first. I love the focus on character development, in a way I never saw (and still do not see) in American superhero comics, which I collected for many years.
Manga and anime for children and tweens often focus on friendship, teamwork and the ability to overcome obstacles and grow stronger. This appeals to people who are not ever going to be the alpha wolf. I may not have rubber limbs, a la Luffy or Mr. Fantastic, but if we work together we'll be the strongest. Rawr.
And lastly, but probably most importantly, manga and anime explore themes that cannot be discussed at the dinner table. Sequential art and animation is a valid and acceptable space for acting out. Of course sex and violence are sometimes part of manga and anime, as they are part of written literature. They are a part of human culture. Talking about a thing, allowing people to express a thing artistically, is a critical component of human culture. We need to draw on cave walls.
Some people like the art, some like the stories, some the themes, some the emotional impact. Some people like to see their own experiences and/or fantasies expressed in a visual medium.
That's why some people like us love anime and manga.