When you open the app on your Apple Watch, you'll be asked if you want to start a run, walk, cycle or "other" workout. Once you give the Runkeeper app permissions to access the data stored in the iPhone's central Health app, you're good to go -- the Watch app can track time, distance, and minute per mile as well as your heart rate data for the entire run.
What's not clear yet is how the Runkeeper app will track your route. One of the iPhone app's main features is the ability to see exactly what route you take as well as the elevation you climb throughout your run -- that all comes thanks to the GPS in the phone as well as its motion processor. While the Apple Watch can certainly track steps and distance on its own, it doesn't seem possible for the Runkeeper app to show you the route you take when using it just with the Apple Watch.
Still, if you're more interested in just tracking your distance and run stats and don't need to know the exact route, Runkeeper's Apple Watch app should provide another good option compared to the built-in Apple Workout app. Still, serious runners should use caution: our testing showed that the Apple Watch's distance tracking features were a bit hit-or-miss, even when you carry your iPhone with you. Going without your phone reduced its accuracy even more, so take these readings with the appropriate grain of salt. We've reached out to Runkeeper to get more details on the new app -- and if we take it for a run, we'll let you know how that goes as well.
Update: We've confirmed with Runkeeper that the lack of GPS means you won't see your routes when using the Apple Watch app without your iPhone. It sounds like Runkeeper's app uses the same motion / fitness calibration methodology that Apple does for its own workout app, which means you'll see better results if you calibrate it by doing some workouts with your phone first before you go Watch-only. If you're addicted to that route-mapping info, though, you'll want to keep bringing your phone along for your runs.