Live started as application for live performances. I never did live performance, but I can imagine you need everything as quick and as agile as possible. With that in mind, they developed pretty unique app that was reliable and well suited for this purpose. (Thanx to it's clever session view thing.)
I believe it was around version 5 or 6 when people started to use Live for classic production as well. I guess main reason was unification of user interfaces used inside and outside of their studios. Around version 7, Ableton implemented better EQ, more cool devices and suddenly people found out, that it's in fact really good and solid DAW.
That quickness of performance tool just reminded. Everything is right under your fingertips and pretty much everything happens in one single window. That's what created Lives amazingly quick workflow. For example setting sidechain compression is question of 3 mouseclicks.
Next thing, that wows you, is its modularity. It's not pure modular DAW, but it has this racking option, that makes it really easy for you to get creative. Wanna layer X synths in one track? No problem. Wanna have several effects on one track working in parallel? No problem. There is not much of possible craziness you couldn't do with it.
Drum racks are amazing too. It's so logical. One track on outside and new track inside of it for every single hit. Easy, clear. Group tracks (available from version 8 I think) allows you to buss several of your tracks completely. If you need some conventional bussing or effect tracks, classic sends also available. ...and everything still happening inside of the one very window.
Unfortunately version 8 went a bit wrong as well. It's great version, best yet, but it has been around for three years or so. Other DAW developers worked hard on new versions. Some functions like vector automation curves, 64-bit instruction code support, or creative stuff like integrated pitch correction are painfully missing in Live. It's great on it's own now, but development should speed up a bit. So point down for that.
Anyway, I'm using Live as my one and only DAW for few years and having no intentions to change that. May be Bitwig can shake this opinion a bit, but that's question of future. Lot of DAWs claim they're quickest way from your brain to the finished track, but imho Live is only DAW you can really agree on that.