Why do anime "spell everything out" for the audience?

This happens more often in series than one-shot movies, and is common in more than just anime. Basically, it's to help the episode stand better on it's own for viewers who are just joining the show, or missed an episode or two earlier, or who have simply forgotten a plot point from an episode two months earlier. This over explaining becomes even more noticeable when watching several episodes back-to-back of a series that was originally aired with days or weeks in between episodes.

Most anime, including Death Note, are adapted from manga (basically the Japanese equivalent of comic books). This leads to a lot of internal narration and recapping for two reasons:

1. Manga, like comic books, are a quite limited medium. In a lot of comic books, particularly older ones, you will see characters internally-monologuing or repeating unnecessary information to make it clear what's happening in the panel, and what the characters are thinking. When manga get adapted into anime, these monologues tend to get included because...
2. Manga are often adapted at a rate of one or two chapters per episode, but if there isn't enough material for a full twenty-minute episode, they're forced to pad it out with filler. This can include stretched-out camera angles, but also egregious flashbacks/recaps and lengthy monologues. My Hero Academia, IIRC, is adapted at a rate of half a manga chapter per episode; take a shot every time Deku flashes back to something you've already seen, and you'll be dead by the end of Series 1.

As you noted, not all anime do this. Some adapt more chapters per episode - comedies and slice-of-life shows do this in particular, as they don't have to worry about pacing. Sometimes new material will be inserted to fill in the gaps, including entire arcs full of filler if they've caught up to where the manga is, but said filler might also contain large amounts of monologues a) so it matches the rest of the series and b) to give the manga an easier time catching up.

tl;dr: Anime is expensive, and often made to make the original material more visible. Repetitive vocal exposition is a handy cost-saving technique.

I want to expand on F1Krazy's answer a bit. As he says, one likely reason is that they need vocal exposition to pad a show's runtime. But then you could wonder "Why is this not done in Western media?" And sure, there have been many western TV shows adapted from other media, and in general they don't seem to use exposition as a crutch as much. I think there are two main differences:

1. Western producers are more willing to take liberties with the original content either out of creativity or to match some current trend.
2. They have a larger budget to play with because the expected revenue is higher.

It's really the second point I want to expand on because anime is a bizarre industry if you take a look at the economics. Often, the point of an anime adaptation is to bring light to the original material, whether that's a manga, light novel, web-comic, or something else. In other words, they serve as high-profile marketing material. They sometimes operate at a loss just because the original work becomes significantly more popular as a result. Why? Profit margins.

Manga and light novels often have fairly high profit margins because it only takes one person or a small group to create the work. Anime, on the other hand, is expensive and time-consuming. The profit margins are extremely slim. But, they have very high visibility. So the show itself does not need to succeed. As long as it significantly improves the sales of the original material, the show has done its job.

Since it's easier and cheaper to pad episodes with vocal exposition (less motion = less work, redundant material), it gets used often as a cost-saving measure, among other techniques. If the show itself manages to become popular enough to be profitable, then maybe they'll keep going. But that's difficult. Because of how tight the margins are, even a show that's doing well on its own might get pulled just because it's not doing well enough to be worth the effort.

Unfortunately, this is just an example of lazy writing. It is much more challenging, and a lot more satisfying for author and reader, to show, rather than tell. It is a lot more fun to leave clues for the reader, or viewer, to find. But this requires a lot of work and practice on the part of the author or scriptwriter. A story that shows you things, instead of simply telling them, is a lot more difficult to do well. It's a rare art these days.