I don't see anyone ever becoming "the netflix of podcasts"... podcasts are more akin to the content on YouTube or Facebook or some other social network.
We don't really want anyone to become the 'Netflix of podcasting' if that means podcast content locked up within their paid service. Podcasts are more like websites... point your browser and go (or in this case, podcast player of choice).
The fear is that Spotify is trying to disrupt that model. I don't know if that is a legit fear or not. Maybe they just intend to to distribute podcasts via their player/service, and then also have a few really high-end paid shows, too. I guess that's their prerogative, just like a few 'traditional' podcasters have some paid content.
But, on the whole, the paid thing runs counter to the podcast culture. Most podcasters who make money, do so via sponsors or 'value for value' contributions from the listeners. Some have tried ads, but I don't know how successful that is, and most listeners kind of hate it.
Why are they “going after Apple”? How big is Apple; and how big are the other players?
Well, Apple is the big 'podcast directory service' (like a phone book listing or Google page result) out there, as well as the top audio player. Apple's Podcast app and iTunes accounted for about 63% of podcast plays last month. Spotify has been growing, and hit about 9.5% last month. For some other perspective, Overcast hit 2.9% and Google Podcast hit 1.1%.
BUT, it is critical to recognize we're just talking about directory and players here. When you create a podcast, you host it somewhere like Libsyn or Blubrry (or somewhere like SoundCloud or Anchor if you want to play it risky). Then, someone uses a player to read your RSS feed to get new episodes. Apple just plays the 'directory listing' role and/or media player role, but really has no control over anything after that. (ie: once someone subscribes with their player, Apple could go away and you'd keep getting episodes.)
My feel on why they (Spotify) are interested, is that podcasting just keeps growing in popularity/use. They want to have some attractive shows to pull people into their subscription service, just like Apple is creating TV shows to hook people into theirs.
Competition should help spur both Apple and Spotify to improve and make their respective podcasting services better.
Maybe, though I'm not sure that's the point here, and that they are really competing. Apple has been doing the podcast thing all along, but for what? I guess recognition. Maybe it has been a selling point for Apple devices and 'iPods' (now phones) to become more popular.
Spotify seems to have other motives, like pulling people into their service (paid). We'll have to watch where this goes.
My first thought is why would I subscribe to Spotify to listen to a podcast. I then see people say they have premium and there are no commercials. I really hate commercials in podcasts. Every podcast has the same mattress or squaretrade or happy meal or whatever commercials. I think I'd pay $10 a month if all podcasts I listen to were commercial free.
Heh, you may just be listening to the wrong podcasts, then.

That said, more and more podcasters take on sponsors, I guess, so you get some ad-reads. But, the better podcasters actually use and love the products from their sponsors, and those products are a great fit for their listeners. If the 'ad read' is just the host talking about why they love XYZ product, that really isn't too irritating. I've bought a number of products over the last few years based on podcast host recommendation, and I've been quite happy to have discovered those products/services. It's not all bad, IF well done.
But, there are also many podcasters who are moving towards a 'value for value' funding model, where they just ask the listeners to donate what the podcast is worth to them. Sure, you probably have to listen to an ask for said funding from time to time, but I really like that model too, as it keeps the podcaster not being influenced by advertisers/sponsors... especially if the podcast is of a type where such influence matters (ie: politics, medical advice, etc, etc.).