Good UI, cloud based, no servers, slick UI, integrated (good) chat, invite links, solid quality, and free.
Mumble is a solid product but there's a lot more friction. First you need to host a server, and for most people that means paying for it. And then to get people to join they have to install a client, and the website can be confusing, and it's not clear which version they need to install, and it keeps prompting you to update which requires it to restart, and the client's UI is clunky, and the integrated text chat is very poor. And then when I need to join someone else's Mumble they have to tell me the host...and the port, and I've got to copy and paste it and go through Mumble's annoying UI to add another server, etc., etc.
Plus Discord's chat is good enough that it makes sense to start building small communities around it; it's also competing with Twitter group chats, Facebook groups, etc. I play World of Warcraft, and some guilds have started using Discord as their main "hub", instead of forums or whatever. And then there's the bots, and the emoji support, and so on, and so forth. Mumble competes pretty well with the core of Discord, but Discord does more than Mumble.
Ultimately, I think it's the same "why did Slack get so big when it's just IRC with some of the irritations filed off and a slightly better UI" question all over again. If you take a product people need, and make it very slightly better and really easy to adopt, then it's not surprising that it will get huge. Yes, you could technically use IRC instead of Slack, but there's a reason why a ton of companies adopted Slack but never adopted IRC.