IMHO, sublime is a lazy yet flexible editor, it has very fancy out-of-box experience, you don't have to download a lot of stuff from the internet and make your own customizations before you can finally gets started with what you intended to do few hours earlier.
I know that both vim/emacs have very handy package manager, but they don't ship with vim/emacs.
I give ST a try for a few months a year or so ago. Maybe this is no longer true, but at the time, the first thing everyone did with ST is download and install Package Control, which is no more integrated with ST and doesn't really work any better than the Emacs equivalent (can't speak to Vim).
My impression was ST gives you something that is almost as powerful as Emacs or Vim without requiring nearly as big of a shift in how you deal with text in the other applications you use all the time. That and the fact that it looks prettier and more like a platform-native application I think explains the appeal. The amount of customization needed didn't seem all that much less to me.
I've found Package Control to be pretty painless. I can go from "Huh, I need syntax support for this language" back to editing in a few seconds.
The other thing is that when I do want to customise something in ST, it will usually just involve editing a single obvious value in a JSON file, rather than hunting through docs to find out what keywords I needed to add to my vimrc.
I know that both vim/emacs have very handy package manager, but they don't ship with vim/emacs.