Why does the male budgie sit in front of the breeding box on the perch all the time while the female climbs around it?

The main reason is that the male is simply curious. he wants to know why the female spends all her time in there. She won't come out to be with him, and being social creatures, budgies love company. So if the female won't come out to the male, the male will simply get as near to the female as he can.