Is there any rationale, or at least historical reason, why many anime end their episodes with spoilers to the next episode? Even those that rely heavily on plot twists, it's very surprising and one needs to stay sharp to skip it early enough (only to discover at the end of a certain anime, that it actually contained additional scenes and not spoilers...). Unlike with "previously on X" recapitulation I see no benefit in this.
It's to get viewers hyped up to see the next episode.
If they do it right, they don't give all of it away, and they leave the viewer with a sense that's there's more to see, and that they need to watch the next episode to figure out what happens or even how it happened.
When anime was first produced, it was very expensive to make and a very time consuming process.
When Osamu Tezuka first created the animated Astroboy, he sold each episode for 750,000 ¥ , even though his estimated cost was 2,500,000¥ per episode. In light of this, anime had to adapt clever ways of saving money.
Tesuka pioneered several cost-saving measures, some of which are still around today. Animating 'on threes' (each celsheet is shown for 3 frames each), recycling cels on other scenes, characters talking but their mouths are not shown and several other items.
After Astroboy's success other companies realised that an anime series was indeed possible despite the seemingly infeasible costs.
Astroboy contained both opening and closing songs, which meant that the studio only had to animate them once, saving roughly 7 minutes per episode. Other studios started to extend these savings with 'Previously On X...' and 'In the next episode of X...'.
So, this is where the origins of 'next on X' lie.
Aside from saving money, The purpose of these scene is to also get the viewers excited to watch the next episode (as @Sam I am has said in his answer).
The problem is, often when shows end on cliffhangers - the entire next episode is how that plot point resolves. Moreso if the studio has reduced animation time per episode to its max, then there isn't much content to pick from for the preview. As previews are sections of already animated material, some parts may not be ready for viewing yet and also likely won't have gone through the rigorous storyboarding and preparatory work that the main feature has.
You can see this in several studios where they've actually made a good effort to not show anything explicitly plot revealing, but the soundtrack has (from being associated with a character).
Something which studios do also is use their animation budget to emphasize key scenes - More attention to detail will be made for a character's death, than when the character has a casual stroll along a pavement. This is a sensible decision by the studios - viewers will remember the amazing transformation scenes that the main character's robot performed, but not so much the animation of a heavy dialogue scene - It will seem to be of an overall higher standard.
On the preview side of this, in order to attract repeat viewers, studios want to pick the most attractive scenes - and yes, they're often the ones that are most important.