Specializations are far more distinguished than they have ever been before in 7.0, and that means that spending the first ten levels without a spec, simply using the "generic" class abilities that all specs use (or replace with something) doesn't really work anymore. For example, there's no more Frostfire Bolt, so Mages really have to have either Arcane Blast, Frostbolt, or Fireball to start off with at level 1.
Reaching level 10 lets you switch specs, and from that point on, you'll have all three (or four if you're a druid, or two if you're a Demon Hunter, though they'll never be lower than level 98) specs available to you any time you're outside of combat.
So what specs do the various classes start with? Generally I think the theme is which spec is closest to the heart of what the class is. With the exception of the Priest, they're all DPS specs, which makes a lot of sense given that you're not going to be doing any group content before level 10. They generally go with a mix of iconic, simple, and flavor.
Assassination and Subtlety would both be decent contenders here, but I think that because Assassination's mechanics are simpler (and since you can't get Gloomblade until level 15, you don't have to deal with backstab) and also draw the most from the classic rogue toolkit. And since most Rogues use daggers, it's got to be a dagger spec.
While Warlocks have moved away from all specs maintaining a million DoTs, Affliction retains that gameplay. It's also pretty simple - the spec is entirely built around maintaining these DoTs. Once again, Affliction has the most old-school abilities.
Arms doesn't have the unique Titan's Grip aspects of Fury nor the tanking role of Protection. It's really the most classic "I'm a soldier who hits things with my big weapon" spec in the game, and is easy to wrap your head around.
Retribution is the Paladin's only DPS spec, and arguably fits with the icon of the Warcraft Paladin - wielding a big ol' warhammer - than the other two specs (I know some Holy Paladins are complaining that the Silver Hand looks like a two-hander, but that just makes them look more like Uther and his guys, which I think is a positive boon.)
Hunter: Beast Mastery
I might have expected Marksmanship, but especially since Marks was conceived in Legion as being always petless - something they thankfully walked back and made a talent - it makes sense that Beast Mastery would be the default choice, as it retains the ranged role and the pets of the classic Hunter archetype. It's also simpler to play than Marksmanship.
This is the only non-DPS spec that you start with at level 1, but that makes sense for a couple of reasons. First off, Priests are the only class with multiple healing specs. Shadow is the minority when it comes to roles. Additionally, Discipline in Legion really blends Shadow and Light to be the "Grey" spec, on top of also blending healing and damage - the latter meaning that the spec should do fine soloing, especially in those early levels. I wouldn't be surprised if you don't even get a heal until after level 10. If they started off Shadow, it would make Priests in general look like a "bad guy class," which healing priests, especially Holy ones, really shouldn't.
This one's not too much of a shock. Elemental is the caster DPS spec, which makes it similar to Restoration (caster, ranged) and similar to Enhancement (DPS.) Lightning Bolt, probably the most iconic Shaman spell, is central to its rotation, so there you have it.
Arguably Arcane is the most flavorfully "Mage" spec, but Frost has always been a very solid choice for soloing, thanks to the slowing effect of Frostbolt. Frankly, all three specs are pretty even on the "iconic" scale, but for ease of play for new players, Frost is probably the best option.
Technically I'd say Brewmaster is the most iconic Monk spec, given that it's based largely on Chen's Brewmaster hero in Warcraft III. But with Windwalker as the only Monk DPS spec, this was probably the right way to go.
Druids are a crazy class, and with four specs, there's no majority on the "ranged caster" and "melee physical" question. But if you were to describe the most notable thing about Druids to someone who had never played the game, you'd probably start with the shapeshifting, and since Balance's shapeshift form is more of a bonus that can come later, Feral seems the obvious choice.
Death Knight: Unholy
Death Knights are pretty famous for raising the dead, which already puts Unholy in a good position. Then you factor in the fact that it's the DPS spec that uses a two-handed weapon, which is the loadout of 2/3 of the DK specs, and the fact that, well, it's a DPS spec, and it's again pretty clear that this makes the most sense. If two-handed Frost were still a thing, it might have had a chance (it is the simpler of the two specs,) but Unholy really nails the iconic factor.
(Fun fact, DKs are the only available class that begins in a rested area, thus not forcing me to wait 20 seconds to log out!)
Demon Hunters: Havoc
Obviously I haven't tried this out myself, but I know that Demon Hunters start off as Havoc thanks to various YouTube videos. It also makes perfect sense - Demon Hunters only have one tank and one DPS spec, so there you have it.
I haven't played any of these guys except the Rogue past level 10 (and he just hit 11,) so I don't know what they've done to adjust when you get your various abilities.
It's a tough balance to strike, making sure that low-level players feel like they have enough stuff going on that they're having fun while also ensuring that leveling up in the higher levels feels like something more than just "oh, and now all my numbers have gone up slightly." This is a problem that will get tougher and tougher as the level cap rises, though I'm hoping that if Legion's zone-scaling works, they can really accelerate the rate at which one levels up without making people feel like they out-level a zone after two quests.
Especially with character boosts, I imagine that there are far fewer players starting brand-new level 1 characters these days. But I'm glad that Blizzard is at least considering how to accommodate those people in a very different world than the one we started in.
To do a brief little flashback, when I started playing back in late vanilla, all classes had their abilities divided between three tabs in the spellbook, corresponding to spec. Some of these didn't make a huge amount of sense, like Consecration, a damaging ability, being on the "Holy" page of the Paladin spellbook.
But at level 1 you would always start with three abilities - one from each spec-page. If I recall correctly, Paladins started with Devotion Aura (Protection,) Seal of Righteousness (Holy, and it used to last 30 seconds, add holy damage to your attacks, and get consumed by Judgment to deal a burst of holy damage to the target - Seal & Judge was almost the entirety of Paladin gameplay back then) and Judgment (Retribution.) I think Mages started with Fireball, Frost Armor, and Arcane Missiles. Anyway, we've come a long way since then.
Blizzard seems to want to give you something cohesive right off the bat with this new system, even if it means that brand-new players might feel pushed toward one spec over the others because of what's familiar. On the other hand, especially with the gearing changes, it'll be a lot easier for them to switch specs once they get used to the game in general.