How do you get extra hospitals?

You can only have a maximum of 8 hospitals in your base. Like any other resource building, click on an empty tile in your base, and build a hospital.

The best advice I can give is to have all 8 hospitals running at max level (i.e. the level equal to your HQ). That way, you can save as many troops as possible (wounded) if they get attacked. The biggest mistake that rookies make is to have 1 hospital which subsequently leads to them losing their entire army (permanently) because more hospitals equals more wounded troops over permanent losses.