What’s the best way to present a sandbox world to your players?


In my sessions, I let my players play in a sand-box world. There is a main story that I always plan that they could follow, and I have no problems with them doing so.
However, I would like my players to fully realise this is a big, breathing world that they can fully explore however they like.

I have already shown them the map of the whole globe, given some standard lore about important places and told them that they can go wherever they like.
I’m hoping to be able to let go of the main story and let them wander around the world, but I’m afraid they’ll remain passive until I throw some encounters towards them, instead of them looking for adventure.

The party is currently level 6, and I would prefer it most if they would gradually expand their influence over the world.

What would be a good way to achieve this? How can I best present a sand-box world to my party, who’ve never done something similar before?

We’re playing D&D 3.5 if that changes anyone’s answer.