Well.. motivation to explore is going to be subjective to an extent but as Destiny stands when we begin playing it.. there's just no other safe place out there. There is a story to Destiny though, and through that we may make discoveries. Maybe there is a faction within a faction of an Alien race that will ally with us, maybe we meet an entirely new allied race that has a similar home to the Tower, or perhaps we simply push out from The Tower and start once again expanding and settling. Who knows?
But as of launch, it's just The Tower. And we, as Guardians, are Archaeologists and Frontiersmen. We head out and scout, scavenge, and study what we find to try to figure what happened and how we can go about changing the tide in our favor again.
As for why everything is after us.. well they are and they aren't. From what we've seen thus far, the enemy alien races don't seem to like each other either. We, as allies of The Traveler, were targeted specifically by The Darkness though. Some, if not all, of those Alien races seem to serve or worship The Darkness (from what I recall reading in the Grimoire), so it seems like it was a priority to destroy us and probably The Traveler itself. It's all pretty unclear intentionally. We'll learn more as we play.