When limited by obstacle limited weight, this means field limited weight is smaller. When field limited weight is smaller, this means balanced field length is smaller than actual field length. When balanced field length is less than actual field length, this means you have excess runway during lift off. If you use this excess runway length for further acceleration you can achieve a greater V2 after lift off. When you achieve a greater V2, this means you can achieve a greater climb gradient. When you achieve a greater climb gradient rate than required for obstacle clearance, this means you have excess climb gradient. When you have excess climb gradient this means you can start the the take off with a greater weight than before.
Regards.