And why does it require legislation if the courts are willing to reject DADT as unconstitutional?
Because Obama doesn't accept that judgement (and it's not final yet, pending appeal).
I don't understand why Obama supports staying the court order that would nix DADT
Because the idea that Congress can't set the rules for the military is a potentially dangerous idea (and one that most observers consider to be wrong). Just because Congress sets a policy that is wrong-headed and dangerous doesn't mean it's unconstitutional.
I gave an example in a related discussion that I'm not going to bother to find. Suppose that Congress passed a law that, to reduce friendly fire casualties, all uniforms must be made of hunter-orange ripstop nylon and be equipped with flashing LEDs so that they can be easily seen.
This would, of course, be a death sentence to American soldiers,.... but it's not far from the historical example of the famous "red trousers" that the French government dictated for its troops in WWI (in the interests of troop morale and espirit-de-corps(e), of course). Obviously this is a bad policy that reduces American military efficiency.
But it would be equally or more dreadful for a court to step in and say that a soldier's right to free expression trumps the military necessity of wearing a uniform, and therefore Congress has no right to set uniform standards of any type.
Basically, Obama -- and most sensible observers -- are
really worried about the baby-to-bathwater ratio this decision involves.