1) Because we recently switched to Sky TV, from an old and financially crippling contract with UPC/Virgin, and the deal includes Sky Go, which lets me watch past episodes of TV shows on my laptop and smartphone. It's pretty nifty.
2) Because, even though I think the contemporary obsession with superheroes is rather infantile, superhero stories have some virtues. One is that they are usually old-fashioned in both storytelling and morality. They generally eschew the shades of grey and anti-heroes which are so prevalent elsewhere. In terms of narrative, they usually have a simple 'quest' storyline, and don't go in for non-linear storytelling or endless atmosphere-building shots.
For this reason I find myself watching more superhero films (and now, TV) than I would like. I wish there were more good old-fashioned stories outside that genre, but there's not.
Also, superhero movies (like science fiction movies) usually look great.
3) Because Supergirl herself is adorkable, to use a nice portmanteau word coined by TV Tropes. At last, an action heroine who is not angry or bitter or out for revenge against men. It's refreshing. (At first, Supergirl is trying to emerge from Superman's shadow. But that seems fair enough, and she's over it by the second series.)
4) It's true that the show is is quite politically correct. But then, isn't almost all TV these days? Superheroes are always presented as symbolic of 'minorities', and all 'minorities' are presumed to be the same-- whether that minority has to do with skin colour, belief system, sexual behaviour or delusion regarding gender. Aliens (like Supergirl) are regarded with suspicion in the show's fictional world, and this is made the occasion for all kinds of left-wing messages. (Which is not to say that "we should be tolerant of difference" isn't a good moral in itself-- it is, although it can obviously be misapplied, and routinely is.)
The more blatant a piece of fiction's ideology is, the less it bothers me. A couple of years ago I read a young adult's horror novel called Gallows Hill by Lois Duncan. I thoroughly enjoyed it. It was blatantly pro-New Age and anti-conservative Christian, but that didn't bother me-- I even enjoyed how hammy it was about it.
Apparently one of the characters in the Supergirl comic book was made black for the TV show. I don't care about that. I don't care what colour or sex the characters in a story are. I haven't seen the new Ghostbusters movie, but the premise of all-female Ghostbusters is fine by me.
Another of the prominent characters is a lesbian. Well, homosexuality is a feature of our time, and entertainment can't ignore it.
5) I watched the first episode on a plane, and that makes me feel strangely invested in it-- even nostalgic for it. Watching a movie or a TV show on a plane is a very unique experience. (I'm told you can't qualify 'unique'. Phooey!)
Superhero comics were not a feature of my childhood. Not that I didn't read comics. I loved comics. But I read British comics; Battle (war stories), Transformers, Eagle (adventure and science fiction), and Roy of the Rovers (soccer). My knowledge of superheroes was limited to toys, the Adam West Batman TV series (and I had no idea it was supposed to be self-parody) and occasional annuals. (I did have a Spider-Man annual, composed of one long story, which I read quite often).
But the whole superhero phenomenon-- the fact that there were dozens of these characters, maybe even hundreds, and complex fictional worlds devoted to each one-- passed me by.
I can remember, towards the end of my childhood, my mother gave me some American superhero comic-books she'd bought in Dublin city centre-- the first ones I'd read (and the last, come to think of it). They actually disturbed me. There were letters from readers (obviously adults) which compared the storylines to various stories in the news and discussed them like they were Shakespeare plays. This is a bit weird, I thought.
Generally, I think superhero entertainments are OK as long as they don't take themselves too seriously. After that, it does indeed get weird.