Because we live in three-dimensional space, there's no reason for us to run out of room unless we stop building (especially upwards).

The constraint isn't necessarily that people have to leave the planet so much as we need to ensure access to minerals and high industrial capacity.

When we exhaust "nearby" minerals and space transit has advanced enough, colonization will look more attractive.

Will we hit a bottleneck along the way? I don't know. But it would be an awful shame if it were political.