The best one-line summary I have read was that 13th Age was what DnD Next / 5th Edition should have been; without the hate for 4e players, and without the teary apology letter to Pathfinder fans. That is, where many 4e fans have felt 5e has ignored what they enjoyed, 13th Age has kept similar options. It has been made from the houserules and home sessions of Rob Heinsoo (lead designer, 4e DnD) and Jonathan Tweet (lead designer, 3e DnD), but has moved onwards, taking elements from each edition and forming something new (instead of simply rolling back to past editions, ala 'Next').
They have some great tricks, some of which can be found in other 'indie' games, but which would work well and be easily implemented in a 3rd or 4th edition game. The Escalation Die is one I see the most talk about, but after backgrounds, I'm not sure I ever want to return to using skill lists! 4e simplified the skill lists of 3e, but Backgrounds makes them a whole new, easy to use, creature!
There are parts that I am still not too sure about - the "simple" classes (Barbarian, Paladin, Ranger) seem too simple on paper. I am withholding judgement until I have seen more of them in action, but the latter two in particular have the feel of 'not enough choice' mixed with 'choosing from the same pool again'. 4e ran aground of this with their 'essential' fighters. You have a pool of powers, and choose some from them. Then, as you level up, you choose another, from the same pool. Then another. Ultimately, despite you gaining levels, the items you choose are getting worse, for all the best (be that mechanical, flavour, or however you chose them) were taken earlier.
But for every element of doubt (Constitution modifier to HP again? +1 to your constitution has huge differences at higher levels) is outweighted by many elements that are great (choosing middle modifiers for defences, more even scores, class and race getting stat bonus options). Even the whole 'Vancian magic' problem has a simple, if not complete, solution, in that instead of 4e's at-will / encounter / daily slots, each spell slot can be filled with a spell of any kind, thus allowing a purely 3e wizard to be created alongside a wizard matching one from 4e.
When I received my 13th Age book, my daughter (currently 4, at least until the end of the year!) was eager to play this 'new game' with me. So we sat down with the book, and talked about what she wanted her character to be. There were times when her attention wavered ("How To Train Your Dragon" was playing in the background), but by dinner time, we had arrived with this:
Clearly, she was heavily influenced by "The Lord of the Rings" and "The Hobbit" (she's only seen the first bit of The Hobbit, until they leave the Shire, and none of the LOTR movies...but she's played through the LEGO game, and I have a ...reasonable... collection of the LEGO figures).
Her character was "Grandolf, the Hobbit Wizard" (I wasn't going to go into the hobbit / halfling distinction), and it was played by a Sauraman figurine, because she liked his cloak. Through some questions, she allocated her stats (10 / 8 / 16 / 18 / 14 / 12) and noted down her defences and hit points.
Ali decided eagerly that one of her backgrounds was going to be that Grandolf liked to breed spiders from eggs (watching the eggs with a telescope (!) to see when they cracked open, and the babies came out). He even carried a spider familiar!
Strangely, the second background was a lot harder to pry out. Every question as to what else Grandolf had done or likes to do kept coming back to those spiders...but eventually, we worked out that he also is an avid gardener, having the most colourful garden in the village. These things turned out to be equally important, and so both gained a +4 bonus.
She then told me that Grandolf grew up on the clouds, which I quickly tied to the Overworld. It seemed to fit that he should start with 2 points (positive) with the Archmage, and one point (positive) with the high Druid, because of his gardening.
By the time we reached spell choices, I think Ali was wishing she had chosen a simpler character (she did keep eyeing off my Fíli-the-Dwarf-Ranger and his Warg companion. I was making this character alongside her, so that she didn't have to adventure totally alone.). But with some simple drawings, I described the spell options to her, and she chose them - but the only ability she wanted to write down was "Evasive". The rest of the numbers were causing her to get a little distracted.
So now Ali has her character sheet, she has a card with her spells written onto it (made by me this time), and she has some dice - this weekend, I foresee some adventuring happening. And I'm looking forward to it!