I can't even imagine what it might mean for " Stripe, Github, Trello, and Zendesk streams" to be integrated into Slack, or why I might want them to be.
Want to share a gist with somebody? Just send a link to the gist to the chat channel, and people can click it.
Now, if I had to tab between panes across 2, 3, 6 app servers, background workers, and database instances, I just wouldn't bother. It's only useful as background noise. But that's fine, because there are tools that let me easily consolidate all of my logs into one stream.
And that's how I see Slack. Just like I can't get a notification every time I have one exception happen in production, because it would kill my workflow, I can't get a notification every time somebody updates a Trello card or resolves a Zendesk ticket. But what I /can/ do is passively watch the stream: Slack is the consolidated logfile, not for my production servers, but for my company.
Could I configure all that with IRC? Yes. Do I want to set it all up, when Slack lets me OAuth against every single service imaginable with one click? No, I really don't. My time is far more productively spent elsewhere.
For integrations, Github tells us when there's a PR. Yes, there's a clickable hyperlink, but I also get to know a bit more inline. It's additional context you don't get with only a hyperlink. There's less switching contexts when things are inlined ... it's a UX feature. I'd call it a feed of things happening across all of my apps, mixed with the ability to discuss those things in a standard place.
The integrations also give you shortcuts to actions without switching from your "command" line. Sure there are _some_ tools you can install on your local machine, but none make it this easy. Butterfield had the same success with Flickr (which yahoo subsequently destroyed) in making a killer user experience. That was for photos, Slack is for communication.
Having this outside of e-mail keeps my e-mail inbox less cluttered. That's my reason anyways.
Emails are in their own isolated packages. If i get an email saying the build broke, there is no easy way to check the last time it happened. I wouldn't be able to glance at the last few messages and see that the build breaks every time "X" commits or the tests fail every monday. I can't easily get context on what someone was doing when it broke.
With something like slack it's all right there. The last 3 alerts, maybe a few messages from devs quickly explaining what happened (or preemptively saying that the build is going to break, and it's okay), someone taking responsibility and saying that they will handle the fix, etc...
It's just nicer.
At my workplace, everyone is usually always active in chat working out problems and having discussions, so a notification is more likely to be seen by more people sooner in the chat than in an email.