I think the idea behind "getting to know" or "warming up" to a story is a lot like getting to know a person you just met, but applied to a setting, a plot, or fictional character(s). You can't usually ask a story about themselves, you have to wait passively for the story to unfold, find out its quirks and whether it's (in this simile) someone worth knowing.
Some people (like you seem to be) can tell pretty quickly whether they like a story or not and either watch it or don't, but other people (like myself and, I assume, like @niinasalmelin) might see a story, like it in theory, but then it takes us awhile to understand what's going on with the story and whether it's worth investing emotional attachment in. It's always a risk that we might be wasting our time, but sometimes it pays off and that's worth it to us.
As an example, I recently binge watched all of Person of Interest up to this point, and it's functionally, conceptually something that's right up my alley, but I might've given up after a few episodes because the characters were kind of shallow and stiff, then I'd've missed out on growing to really like them as their stories developed later. (I dunno if this is a great example because the writing for Person of Interest is kind of spotty, but I'm definitely invested enough to see it through to the end so that's gotta be in its favor.)
