I was initially confused about this as it's not obvious what puts it into each mode but here's the rule: if you have a Keynote presentation open in Keynote on your iPhone when you launch the Watch app, it will control that presentation. If you're looking at the Keynote file picker when you launch the Watch app, it will go into Keynote Remote Remote mode and start trying to connect to Keynote on your Mac or iPad. That's a lot of Keynote.
I'm fairly cynical about this watch thing, but I can definitely get behind use cases like this. The watch could be a great presentation tool if it starts to integrate vibration responses for timers and slide tranistion alerts. It could make you seem almost omniscient, staring your audience in the eye while changing slides and triggering animations at the exact right time. All of this could be enabled by an ordinary looking watch on your wrist.