What makes them emo?

If you heard a bunch of emo albums in a row, you'd get it more, I think.

Okay, I just read that AMG page, and while they're probably technically accurate, it's a little too encompassing for what most folks these days would refer to as emo. (Personally, I'm not a big fan of pigeonholing things, but I understand the need to draw similarities.)

SDRE is defininitely one of the big names in emo. Most of the music that's called emo these days is going to be musically similar. Usually distorted guitars, almost whiny singing, often almost cloying melodies, but melodies nonetheless, and often very pop-oriented at the core.

Lyrically, it originally involved a lot of depressed teenager diary material, but most of the bands that are still around have gotten past the juvenilia, but still speak about the same sorts of personal reflections. There are some bands considered emo almost solely for the music, though. (That is, the lyrics don't have the same personal quality.) Even some geek rock (Weezer, Nerf Herder, Ozma, etc.) can fall in the emo category.

There have been a few popular acts in recent years that might fit the category, if on the very edges, like Blink-182, Jimmy Eat World, the aforementioned Weezer, uhhh, I feel like there's some I'm missing.

There are also some hardcore bands whose lyrics are emo-like, but the music is not. These can be categorized emocore. Snapcase comes to mind. I supposed I can see how Zen Arcade (referenced in the AMG article) might be a progenitor of this in a way, but only as much as you might call Neil Young grunge.

Really, like most such things, it's hard to define in words. Personally, I'd ignore everything in the article before it mentions SDRE and you'd be closer to reality.