I use an app[0] that pre-downloads them, and I listen to them on the treadmill at the gym; while cooking or doing household chores; as background when I'm coding at night; and, sometimes, to help me get to sleep. For every one of those uses except the treadmill I sometimes listen to music instead.
Oh yeah, and I sometimes listen to current-affairs stuff in the morning, but owing to time zone differences that can be tricky.
I'm sure a lot of people listen to them while driving or otherwise commuting, but my commute is by tram and by foot, and it doesn't fit that well.
I don't find it distracts me from cooking, though from coding sometimes yes (then I switch to music). I don't just sit in my living room listening without anything else going on, but I don't think that would be weird. I'd listen to podcasts in the bath if I had a bathtub.
It took me a while to find a set of podcasts I really like. This is different for everyone, but I think it's worth spending some time on. I'm still looking for good ones on a few topics that seem under... undercasted? I think there's still room for a lot more good podcasts, and some of them will even make a good living for the host.
You can download everything on WiFi if you like, but they're just audio files so it doesn't eat that much out of a normal-ish mobile data plan.
Give them a chance, and try a bunch of different ones before making up your mind. I ended up liking podcasts way more than I expected I would.
[0]: https://overcast.fm