It's partly a matter of targeting specific audiences. Depending on what time a show broadcasts at, you can tell roughly what their target audience is.
Plenty of anime are broadcast in the morning. From this season, Danchi Tomoo airs at 9:30, Dokidoki! Precure at 8:30, Toriko at 9:00, One Piece at 9:30, Uchuu Kyoudai at 7:00. These are mostly shows which are aimed at younger children. Those also all air on weekends. There are some early morning anime which air on weekdays as well, but typically they air before school hours begin.
There is also some anime which broadcasts in the evening. These are generally programs which would be appropriate for a family to watch together or for older children/teenagers to watch on their own. Some examples of this from this season are Uchuu Senkan Yamato 2199 (5:30 p.m.), Train Heros (5:30 p.m.), Doraemon (7:00 p.m.), and Yuugiou! Zexal II (7:30 p.m.).
Late night anime are typically those which are targeting a mostly adult fanbase. In many (but not all) cases they aren't appropriate for younger viewers. Even if they are, they wouldn't be interesting to those viewers. At least in principle, the broadcast time is late at night so that children won't accidentally tune in while adults can still watch it or record it. Late at night, TV stations do not enforce broadcast restrictions as strictly as during primetime hours. Late-night anime thus has the connotation of having a lot of fanservice and moe, though that isn't always the case. For many of the shows, like Chihayafuru, it would probably be fine to show it in the evening, but since there are already established time blocks for showing anime, sometimes shows like that get shown as late-night anime anyway.
It's also true that most late-night slots are usually (though not always) purchased from the TV station by the production company directly. Since this doesn't require external sponsors, and because it's cheaper airtime, it's typically easier to get a late-night slot than a primetime slot, so this has become the default for a majority of anime which are not aimed at children. This is done essentially as a form of advertising for the final product (DVDs). Incidentally, this is also the reason why most anime production companies are gladly willing to sell rights to simulcast for next-to-nothing, because it serves as advertisement outside Japan, but DVD licensing rights are expensive (because the DVD is the final product and the way which they expect to make money).
Late night anime slots have existed on-and-off since the 1960s (one of the earliest being Sennin Buraku from 1963-1964), but the first really successful series was Those Who Hunt Elves in 1997. This was also at the start of an anime boom, and rather than trying to expand in the daytime, many new anime series opted to follow this and go for night slots. In recent years, this has become the default method for broadcasting anime, and a majority of anime are broadcast in late-night slots. Occasionally, some of those which are particularly popular will get rebroadcast in other time slots.
As for why late-night series in other countries sometimes get better slots than in Japan, it's likely because of the different business model. In Japan, late-night TV are essentially advertisements for DVDs. They are not aimed at being profitable on their own. In most other countries, there is a much smaller supply of anime broadcast (some of which was broadcast late-night in Japan), and it is often sponsored or on pay-to-view channels. Hence, they can afford to broadcast a lot of anime during the day because of less competition and better sponsorship.
English Wikipedia has some more information, though the article is poorly sourced and out-of-date. Japanese Wikipedia is a little bit better and more in-depth, but still has sourcing issues.